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\begin{document}

\title{Belief Updating and the Demand for Information}

\author{Sandro Ambuehl and Shengwu Li\footnote{
Ambuehl: Department of Management UTSC and Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, ON, M5S 3E6, Canada. Li: Society of Fellows, Harvard University, 78 Mt. Auburn Street, Cambridge, MA, 02138, USA. We are deeply indebted to our advisors B. Douglas Bernheim, Paul Milgrom, Muriel Niederle, Alvin E. Roth, and Charles Sprenger. We are grateful to Itay Fainmesser, Simon Gaechter, Mark Machina, Jeffrey Naecker, Joel Sobel and participants at the Stanford Behavioral and Experimental Economics Lunch, the conference of the Economic Science Association, the conference of Swiss Economists Abroad, the Bay Area Behavioral and Experimental Economics Workshop, and the Stanford Institute for Theoretical Economics for very helpful comments and discussions. Financial support by the Department of Economics at Stanford University is gratefully acknowledged. This research was approved by the Stanford IRB in protocol 27326.}}

\maketitle

\begin{abstract}

How do individuals value noisy information that guides economic decisions? In our laboratory experiment, we find that individuals underreact to increasing the informativeness of a signal, thus undervalue high-quality information, and that they disproportionately prefer information that may yield certainty. Both biases appear to be  mainly due to non-standard belief updating.  We find that individuals differ consistently in their \emph{responsiveness to information} -- the extent that their beliefs move upon observing signals. Individual parameters of responsiveness to information have explanatory power in two distinct choice environments and are unrelated to proxies for mathematical aptitude.


\end{abstract}

\vspace{0.5cm}
\emph{JEL classification: C91, D01, D03, D83}

\vspace{0.5cm}
\emph{Keywords}: Demand for information, belief updating, responsiveness to information, probability weighting, experimental economics

\newpage
\setstretch{1.2}


\section{Introduction}\setstretch{1.5}

Modern economies increasingly trade not just in physical goods, but in information. Economic agents consider whether to purchase medical tests, consulting expertise, or financial forecasts. Entire industries are premised on the sale of useful but noisy information.

In this paper we use a laboratory experiment to investigate whether, and why, the demand for information systematically deviates from the predictions of the standard rational agent model. In contrast to the existing literature on the demand for information, information in our setting is instrumentally useful  but concerns a state that is of no intrinsic relevance to subjects,  and strategic considerations play no role.

The purchase of information is commonly conceptualized as the decision to partake in a two-stage lottery. In the first stage, the agent updates her beliefs upon observing the realization of an informative signal. In the second stage, she chooses an action, upon which the state of the world is revealed and payoffs are realized.\footnote{This conceptualization of the demand for information has been chosen by a wide range of theoretical papers, including, for instance, \citet{AzrielyLehrer}, \citet{AtheyLevin}, \citet*{Cabralesetal.Appeal}, and \citet*{Cabralesetal.}.} A standard rational agent will update beliefs according to Bayes' rule, and make choices according to expected utility theory.  For any agent who correctly anticipates his own belief updating, this suggests two mechanisms through which empirical behavior could deviate from the predictions of the standard model.

First, the demand for information may be affected by systematic deviations from expected utility theory in compound lotteries. These have been documented and modeled, for instance, by \citet{BernasconiLoomes}, \citet{ErginGul}, \citet{HalevyEllsbergRevisited}, \citet{HalevyFeltkamp}, \citet{Segal}, \citet{Seo}, and \citet{YatesZukowski}. Second, the demand for information may influenced by biases in belief updating (reviewed in \citet{CamererOverview}). Our design allows us to estimate the effect of the latter channel on the demand for information, and to identify three biases in belief updating and the demand for information.\footnote{This exercise, however, does not decompose behavior into beliefs and \emph{preferences}.  In the subjective expected utility framework, preferences are basic, and beliefs are just part of the representation \citep{Savage1954}. To be scrupulous, when we say that a subject deviates from Bayesian belief updating, we mean ``in the representation that rationalizes this choice, the beliefs differ from the objective Bayes posteriors.''}

In our experiment, subjects reveal how valuable they think it is to first observe an informative signal about a binary state of the world and then make a guess about it. At the very end of the experiment, a correct guess may be rewarded with a prize. Since our elicitation is in terms of probability units of winning the prize, our results are independent of the shape of subjects' utility function for money. Discriminating between channels is possible as a second stage asks subjects to reveal posterior beliefs for each of these signals. A third stage presents subjects with a conditionally i.i.d. sequence of signal realizations that depends on the state of the world. It measures how much information subjects require until they first prefer betting on the state of the world to an outside option.


We identify two biases in the demand for information that depend on the properties of \emph{information structures}.  Subjects value signals of varying informativeness as though they were more alike, compared to the predictions of the standard model (the \emph{compression effect}). Additionally, subjects disproportionately prefer information structures that may perfectly reveal the state of the world (the \emph{certainty effect}).  We also test how valuations change if we increase the precision of the signal in one state of the world and decreasing it in another, such that the utility of a standard rational agent is unchanged. A class of theories of non-standard risk preferences predict that this will make an information structure less attractive to subjects \citep{Segal, Seo}. Our data do not support this hypothesis.


We identify a third bias that is \emph{individual-specific}. Subjects in our experiment are heterogenous in \emph{responsiveness to information}, the extent to which their beliefs move upon observing information. Importantly, this heterogeneity is \emph{consistent} within individuals. We measure this individual tendency in one task, and find that  it is significantly correlated with subjects' behavior in two additional tasks.  Moreover, while it is related to cognitive style, it is uncorrelated with proxies for mathematical aptitude.  In particular, neither the direction nor the absolute size of this bias is correlated with whether a subject knows Bayes' law, has taken a statistics class, or has a STEM major.


Following \citet{GretherQJE} and \citet{HoltSmith}, we propose a one-parameter model of responsiveness to information and estimate each subjects' parameter using data only from the latter part of our experiment.  This procedure collapses all the data about each subject's belief updating to a single parameter, yet it explains 80\% of the individual variation in the demand for information that can be explained with a model that uses each individual's beliefs data in a maximally flexible way.  Our individual-level estimates of responsiveness to information  are also significantly related to subjects' behavior   in our environment that gradually uncovers information and reveals how quickly subjects' beliefs about the state of the world cross a given threshold.\footnote{ We emphasize that heterogeneous responsiveness to information is not observationally equivalent to heterogeneous risk aversion. To see this, consider an agent who chooses between a safe option and a risky option, and who purchases information that might affect the optimal decision.  If, in the absence of information, the agent would choose the risky option, then his willingness to pay for information is \emph{increasing} in risk aversion.  If, in the absence of information, the agent would choose the safe option, then his willingness to pay for information is \emph{decreasing} in risk aversion.  By contrast, his willingness to pay for information is \emph{increasing} in responsiveness in both of these cases. (Appendix \ref{model} formalizes this argument.)}


Conditional on the estimated parameters, the responsiveness model also generates the compression effect and the certainty effect. This is perhaps surprising, since the model is set up to capture a single individual-specific bias, while these two effects pertain to properties of the information structures.  Both of these are consequences of the fact that our average subject is less responsive to information than a Bayesian, but correctly interprets perfectly informative and perfectly uninformative signal realizations. Formally, the model of responsiveness to information is equivalent to applying a prospect theory probability-weighting function to the Bayesian posterior. The information structure-specific effects we observe are implications of the inverse S-shape of that function.


 Many of our results could in principle be due to either non-Bayesian belief updating, or due to non-standard risk preferences. We argue that the latter are unlikely to explain  the entirety of  our data, for three reasons. First, by design, our results cannot be due to the shape of subjects' utility function for money. Each choice in our experiment elicits, for some event $X$, the value of $k$ such that the subject is indifferent between \emph{receiving \$35 if with probability $k$ and \$0 otherwise}, and \emph{receiving \$35 if event $X$ occurs and \$0 otherwise}.\footnote{This mechanism was suggested in \citet{Allen}, \cite{Grether92}, and \citet{Karni}, \citet{SchlagVanDeWeele}, and has been used, amongst others, by \citet{HoelzlRustichini} and \citet{Moebiusetal.}.}  For any expected utility maximizer, the value of $k$ that leaves such an agent indifferent does not depend on his degree of risk aversion  \citep{Roth1979}.


Second, also by design, they cannot be due ambiguity aversion, at least if interpreted in the standard fashion. In our experiment, all probabilities are objectively given. Subjects have induced priors about the state space, and all information structures are explicitly given, so that the probability of any payoff-relevant event can be calculated, via a simple application of Bayes' rule.  In contrast, ambiguity aversion models are usually interpreted to apply under Knightian uncertainty, where objective probabilities are not applicable \citep{Knight1921, Ellsberg1961, BaillonBleichrodt2015}.


Third, there are other accounts of decision making under risk that are not excluded by our design choices. These do not account for the \emph{compression effect} in the valuation of information structures, as this effect vanishes once we account for elicited belief updating data. The extent of the \emph{boundary} effect is affected by controlling for beliefs data, but some of it remains even then. This suggests that it is partially rooted in risk preferences.\footnote{ We do not have data, however, to allow us to make claims about the extent of \emph{individual-level} heterogeneity that is due to non-standard belief updating as opposed to non-standard risk preferences. Therefore, we make no claims about the extent of total individual-level variation our measure of responsiveness captures; but merely that it captures the bulk of that part of the individual-level variation \emph{that can maximally be explained by non-standard belief updating}.}


Our study contributes to the literature in several ways. While responsiveness to information relates to both base-rate neglect (\citet{KahnemanTverskyRepresentativeness}) and conservatism (see \citet{PetersonBeach} for a review), we show that it is \emph{heterogenous} across subjects, \emph{stable} within subjects, and has explanatory power across three choice environments.\footnote{The literature on belief updating is vast, and we do not attempt to review it here. See \citet{CamererOverview} for a review. \citet*{RabinNBLLN} contains a recent comprehensive review of belief updating experiments that use binary states and binary signals.} We explicitly relate biases in belief updating to prospect theory probability weighting (as reviewed, for instance, in \citet{EpperFehrDuda}).


A handful of papers have studied individual heterogeneity in belief updating.  Most closely related is the successor to our paper by \citet*{Buseretal.Updating} who also find, in a setting with ego-relevant beliefs, that responsiveness to feedback is a personal trait that predicts selection into competitive environments. (They do not relate the trait to information demand, and do not consider variation in information structures).  \citet{AugenblickRabin} show that if a forecaster's belief stream for a given geopolitical event is relatively volatile, then the same forecaster's belief stream for other events is also relatively volatile.  \citet{ElGamalGrether} fit a finite mixture model to binary-choice data, and find that the most frequently used updating rules are Bayes, base-rate neglect  (which roughly corresponds to excessive responsiveness), and conservatism (which is equivalent to insufficient responsiveness).  These authors do not study individual consistency across different kinds of updating tasks.\footnote{Individual heterogeneity in belief updating was observed (but not explicitly investigated) in \citet*{PetersonSchneiderMiller}. \citet*{Moebiusetal.} also find heterogeneity in belief updating in an ego-relevant task, and show that beliefs are equally predictive of subsequent behavior separately for the more and less conservative half of subjects.} Our paper complements \citet*{Antoniouetal.} who document individual heterogeneity in how strongly individuals react to the salience of information, keeping constant its diagnosticity.

Within the empirical literature on the demand for information, our study contributes to the literature by investigating biases pertaining to different information structures (characterized by their state-dependent signal accuracies), and by relating them to a stable individual trait that characterizes belief updating. These results distinguish it from \citet{Hoffman} who studies internet-domain-name traders in a lab-in-the-field experiment. He finds that the willingness to pay (WTP) for information  increases with a subjects' uncertainty about the state of the world, increases with the accuracy of the signal, as predicted by the Bayesian benchmark, but does so to a lesser extent than predicted by theory. This is related to the compression effect we document in the demand for information.


A related literature documents that subjects do not always sufficiently account for the properties of the data generating process when making inferences (\citet{MasseyWu}, \citet{FiedlerBook}). Our finding that subjects' are willing to pay too much for low-quality information, and not willing to pay enough for high-quality information complements \citet{FalkZimmermann} who find that in a context of \emph{non}-instrumental information, subjects prefer few realizations of high-quality information to many realizations of low-quality information.\footnote{Our work may appear related to a literature on decision making under ambiguity \citep{GilboaMarinacci}. We do not interpret our data in light of this literature, for two reasons. First, our experiment involves only objective uncertainty. Second, the literature of decision making under ambiguity has not yet provided a canonical model of belief updating. In contrast, there is a long tradition of analyzing belief updating with objective probabilities (e.g. \citet{PetersonBeach}), as well as on deviations from expected utility theory when objective probabilities are given (e.g. \citet{KahnemanTversky1979}). Our analysis is closely related to these strands of literature.}


We know of no further  empirical  work on the demand for information in a non-strategic setting in which information is instrumentally useful  but concerns a state that is of no intrinsic relevance to subjects,  and strategic considerations play no role. Outside of our domain, \citet{EliazSchotter} show that subjects are willing to pay for information that will not change their decision, if it decreases their uncertainty about having taken the right action, and \citet{BastardiShafir} find that once subjects decide to purchase information, they base their decisions on it, even if they would not do so if they were given the information for free.  \citet{Masatliogluetal.} study how the demand for non-instrumental information varies with the properties of the available information structures. \citet*{Burksetal.}, \citet{EilRao}, and \citet*{Moebiusetal.} find that when beliefs are ego-relevant, WTP for information is larger the more confident the subject is that information will be desirable. \citet*{Kocheretal.}, \citet{FalkZimmermann}, and \citet{Zimmermann} study preferences over the dynamic arrival of information. The first of these papers finds that subjects prefer later resolution of uncertainty about a lottery ticket, the second finds that they prefer later resolution about whether an unpleasant event will occur if they are not forced to focus on that event, and the third that subjects display no preference over clumped vs. piecewise information regarding a lottery ticket. The demand for dynamic information is also studied in the literature on sequential search (for instance \citet*{Schotteretal.}, \citet*{Caplinetal.}, and \citet*{Gabaixetal.}) and in the literature on explore-exploit dilemmas (see \citet{ErevHaruvy} for a recent review.) Finally, endogenous information acquisition has been studied in a wide range of strategic settings. For example, \citet{KublerWeizsacker} focus on social learning and informational cascades, \citet{SzkupTrevino} on global games,  \citet{GretschkoRajko} on auctions, and \citet{Sunder} and \citet{CopelandFriedman} on experimental asset markets.


The remainder of this paper proceeds as follows. Section \ref{design} presents the experimental design and our hypotheses. Section \ref{AggregateResults} analyzes behavior as a function of the information structure, documents the compression and certainty effects, and argues that they are due to belief updating biases rather than due to non-standard risk preferences. Section \ref{Heterogeneity} studies individual-level consistency, presents our results regarding responsiveness to information, and links them to the information-structure specific biases. Finally, section \ref{conclusion} discusses potential applications of our findings and concludes.


\section{Experimental Design}\label{design}


We present a brief overview of the design before discussing specifics and hypotheses below.

We consider a setting in which information is valuable because it informs a subsequent choice. We focus on the \emph{prediction game}, which is arguably the simplest nontrivial such setting. The state of the world $s$ is either 0 or 1. An agent with some prior belief observes an informative signal about the state, and guesses the state of the world. If (and only if) his guess is correct, he receives a prize $k$.

We are interested in how agents value such an informative signal, and how this depends on its properties. Hence, we elicit the probability $v$ such that the agent is indifferent between \textit{receiving the prize with probability $v$} and \textit{playing the prediction game}.  This is the \emph{information valuation task}.

If these valuations differ from those a Bayesian expected utility maximizer would have, we would like to pinpoint the source of the deviation. The experiment thus separately examines each of the steps a rational agent would make to decide about her valuation. These involve  calculating the posterior belief she would obtain and the action she would take, for each possible signal realization, on the one hand, and calculating the likelihood of observing each signal, on the other. To arrive at the valuation $v_I$, the Bayesian then simply aggregates these probabilities. Accordingly, we elicit subjective posterior beliefs (the \emph{belief updating task}), as well as subjective probabilities of observing each signal (the \emph{probability assessment task}). We elicit these elements only \emph{after} eliciting valuations of information structures, so we do not nudge the subjects towards thinking about information valuation in a specific fashion.

Finally, if we find that that empirical belief updating behavior is non-Bayesian, we would like to study the extent to which these deviations are systematic and stable within individuals. The experiment ends with the \emph{gradual information task} which allows us to answer this question by testing how closely deviations in one task are associated with deviations in another task.

We now discuss our setup in detail, outline our hypotheses, and describe how we implement the experiment.


\subsection{Formal setup and hypotheses}


\paragraph{Setup}
We use $s \in \{0,1\}$ to denote the state of the world, about which subjects have an objective prior belief $P(s=1) = \frac{1}{2}$. We let $\sigma \in \{0,1\}$ denote is the realization of a stochastic signal the subject may observe. For each such signal, subjects are aware of the pair of state-dependent probabilities $P_I(\sigma=1|s=1)$ and $P_I(\sigma=0|s=0)$. We refer to such a pair $I = \big(P_I(\sigma = 1|s=1),P_I(\sigma = 0|s=0)\big)$ as an \emph{information structure}.  Notice that, even though both states are equally likely, both signals need not be equally likely.

We let $v_{I,i}$ denote the probability that renders subject $i$ indifferent between playing the prediction game with information structure $I$, on the one hand, and receiving the prize with objective probability $v_{I,i}$, on the other. We refer to $v_{I,i}$ as agent $i$'s \emph{valuation} of information structure $I$. Because there are only two possible levels of payoff a subject can obtain (she wins the prize or does not), this measure of valuation is independent of the curvature a subject's utility function for money.


Observe that \emph{subjective} posterior beliefs $P_{I,i}(s|\sigma)$ and signal probability assessments $P_{I,i}(\sigma)$ are sufficient to obtain the valuation of information structure $I$ for any agent who reduces compound lotteries, has preferences that depend only on final outcomes, and strictly prefers obtaining the prize $k$ to not obtaining it--- even if he is not Bayesian. Given a strategy $g(\sigma)$ that maps a signal realization into a guess about the state of the world, $i$'s valuation of information structure $I$ is given by
\bmath
	v_{I,i} = {P}_{I,i}\big(g(\sigma)|\sigma = 1\big){P}_{I,i}(\sigma = 1) + {P}_{I,i}\big(g(\sigma)|\sigma = 0\big){P}_{I,i}(\sigma = 0).
\emath
Our design imposes $P_I(\sigma = 1|s=1) > 0.5$ and $P_I(\sigma = 0|s=0) > 0.5$, so that the optimal guess is $g(1) = 1$ and $g(0) = 0$.  Then the above equation reduces to:
\bmath \label{ValuationEqn}
	v_{I,i} = {P}_{I,i}\big(s=1|\sigma = 1\big){P}_{I,i}(\sigma = 1) + {P}_{I,i}\big(s=0|\sigma = 0\big){P}_{I,i}(\sigma = 0).
\emath

Our experiment elicits each constituent part of equation (\ref{ValuationEqn}). The \emph{information valuation task} elicits $v_{I,i}$, and the
\emph{belief updating} and \emph{probability assessment} tasks elicit ${P}_{I,i}(s|\sigma)$ and $P_{I,i}(\sigma)$, respectively, for each possible signal realization $\sigma \in \{0,1\}$.

Finally, in the \emph{gradual information task}, we present subjects with a different kind of belief updating problem  to further test whether subjects' responsiveness to information has explanatory power  in other decision environments. Subjects are shown a sequence of conditionally independent signals that are informative about the state of the world, and decide, after each signal, whether they prefer to take part in a lottery with an explicitly given winning probability, or whether they prefer to bet on the state of the world.


\paragraph{Hypotheses}\label{hypotheses}

In our setting,  the space of all information structures is the unit square $[0,1]\times[0,1]$, which we can display graphically. We focus on the subspace $[\frac{1}{2}, 1] \times [\frac{1}{2}, 1]$ on which following the signal is the optimal guessing strategy, $g(\sigma)=\sigma$.


How would a Bayesian agent value the prediction game with an information structure $I$? By replacing ${P}_{I,i}(s|\sigma)$ and ${P}_{I,i}(\sigma)$ in equation (\ref{ValuationEqn}) by their objective counterparts, we derive
\bmath \label{v_I_theor}
v_I^{Bayes} &=&  \frac{P_I(\sigma = 1|s=1) + P_I(\sigma = 0|s=0)}{2} \label{v_theor}
\emath

 Figure \ref{idc} (A) plots the level curves of $v_I^{Bayes}$ over the set of information structures under consideration. We call $v_I^{Bayes}$ the \emph{informativeness} of information structure $I$; it is the probability that the signal matches the state of the world, and it increases to the northeast of the figure.

We study information structure-specific biases by answering the following questions. 1. Does subjects' valuation of information increase one-for-one with with the Bayesian valuation as we move an information structure along the diagonal, or do subjects under- or overreact to such a change?  2. Do subjects have disproportionately high or low valuations for information structures on the northern or eastern boundary of figure \ref{idc} (A), which provide certainty for at least one of the signal realizations? 3. Are the empirical indifference curves indeed linear in the interior of $[\frac{1}{2},1]\times[\frac{1}{2},1]$, or do subjects have preferences for (or against) asymmetric information structures, which would cause the indifference curves in figure \ref{idc} (A) to be strictly concave (convex)?  4. Are deviations from the theoretical predictions due to non-Bayesian belief updating, or are they caused by non-standard preferences over two-stage lotteries?


We then focus on individual-specific biases and investigate whether an individual's behavior in simple belief updating tasks is predictive of her behavior in other choice problems.  We fit a single-parameter model using only each subject's behavior in belief updating tasks, and employ that to predict her behavior in information valuation task and in the gradual information task. Finally, we tie this model to the information structure-specific biases we document in the first part of the analysis.


\begin{figure}
\begin{center}
\begin{tabular}{cc}
\hspace{-0.5cm}\includegraphics[width = 0.4\textheight]{../DATA_ANALYSIS/graphs/idc_edited.pdf} \hspace{-2cm}&
\hspace{-1.9cm}\includegraphics[width = 0.4\textheight]{../DATA_ANALYSIS/graphs/infostruct_nonmirror.pdf}\hspace{-1.2cm} \\
(A) & (B)
\end{tabular}
\end{center}
\caption{\label{idc} \label{set_of_info_struct} Panel (A) displays the set $[0.5,1]\times[0.5,1]$ of information structures and the level curves of $v_I^{Bayes}$. We call an information structure $I'$ that lies to the northeast of $I$  more \emph{informative} than $I$. We call an information structure $I'$ that lies to the northwest of $I$ more \emph{asymmetric} than $I$. Panel (B) displays the set of information structures we used in the experiment and the order in which subjects went through them. (Approximately half of the subjects proceeded through the information structures in reverse order.) The horizontal axis measures the proportion of black balls in box $X$ and the vertical axis measures the proportion of white balls in box $Y$.}
\end{figure}


\subsection{Implementation}\label{Implementation}

In most parts of the experiment, subjects face computerized pairs of boxes filled with black and white balls, as in figure \ref{Boxes}. Box X contains 10 balls, the majority of which are black.  Box Y containes 20 balls, the majority of which are white. In many parts of the experiment, the computer first randomly and secretly selects a box, and then draws a ball from the selected box for the subject to see. The box selected corresponds to the state of the world $s$, while the color of the ball drawn corresponds to the signal realization $\sigma$. We chose different ball totals in Box X and Box Y in order to ensure that na\"{i}ve ball-counting heuristics are not observationally equivalent to Bayesianism in the \emph{Information Valuation Task}.\footnote{If there were 10 balls in each box, then the result of the following heuristic would coincide with the Bayesian valuation: Count the number of black balls in box $X$, count the number of white balls in box $Y$, sum them up and divide by 20.  Appendix \ref{appendixUrnSize} shows that this difference in urn sizes does not affect our results. }


We ensure incentive compatibility by paying subjects for a single decision randomly selected from the entire experiment. Subjects never receive feedback about their decisions until the very end of the experiment. Each subject participates in each of the following five parts of the experiment,  in the order listed here.

\begin{figure}
\begin{center}
\includegraphics[width = 0.7\textwidth]{boxes.pdf}
\end{center}
\caption{\label{Boxes} Screenshot from the experimental interface.}
\end{figure}


\paragraph{Part 1: Prediction Game}
Subjects play six rounds of the prediction game with different information structures so that they can familiarize themselves with the game that they are subsequently asked to value.\footnote{ We draw signals independently and randomly for each subject, conditional on state, according to the current information structure.   The information structures used are $\{(0.5, 0.9),(0.7, 1),(0.6,0.6),(1, 0.55),(0.6,0.85),(0.8,0.6)\}$. We use six information structures so that each subject is quite likely to see both black and white draws, but also so that this part of the experiment is concluded quickly.}  In each round, the computer randomly and secretly selects a box, each with 50\% probability, and shows the subject a ball drawn at random from that box.  The subject then guesses which box was selected, receiving \$35 (\$0) for a correct (incorrect) guess.  Subjects play the prediction game for six different information structures.


\paragraph{Part 2: Information Valuation Task}


This part elicits subjects' valuations of information structures $v_{I,i}$. Subjects are presented with the 10 pairs of boxes depicted in figure \ref{set_of_info_struct} (B). We chose them so as to easily test our hypotheses.\footnote{The choice of information structures also reflects power calculations calibrated to the test by \citet{AbrevayaJiang}. After obtaining the data, we discovered that this test is subject to substantive bias with small sample sizes, and hence we do not apply it.} For each pair of boxes, subjects fill in a multiple choice list, choosing between pairs of options: ``Play the Prediction Game'' or ``Win \$35 with chance $p$'', for $p$ between 1\% and 100\%, in $3\%$ intervals. The value $p$ at which a subject switches from preferring the former to the latter option reveals her valuation  $v_{I,i}$ of the Prediction Game with those pairs of boxes.  Since each line counts as a separate decision, one of which might be randomly drawn for being carried out, truthful revelation is strictly optimal. We constrain subjects to have at most one switching point.\footnote{\citet*{AndersenMPL} finds that enforcing a single switching point in a multiple decision list has no systematic effect on subject responses.}   Half the subjects, chosen at random, evaluate the ten information structures in the order indicated in figure \ref{set_of_info_struct} (B), the remaining subjects evaluate them in reverse order.

 If at the end of the experiment a decision from this stage is randomly chosen for implementation, a subject who has chosen to play the prediction game first sees a ball drawn from one of the two boxes, and then decides which box to bet on. If, and only if, the subject's guess coincides with the box from which the ball was drawn does she receive the prize.


\paragraph{Part 3: Belief Updating Task}


We use the strategy method to elicit posterior beliefs $P_{I,i}(s = 0|\sigma = 0)$ and $P_{I,i}(s=1|\sigma = 1)$ for each of the ten information structures in figure \ref{set_of_info_struct} and for each possible signal realization.  Subjects fill in a choice list to be implemented if the drawn ball is black, and a choice list to be implemented if the drawn ball is white.  These consists of the lines ``Receive \$35 if the Selected Box is Box X'' and ``Receive \$35 with chance $p$'', for $p$ between 1\% and 100\%, in $3\%$ intervals.  Subjects know that if a round from this is drawn for payment, a ball will be drawn from the selected box at random, and their decision on a randomly chosen line of the corresponding choice list will be implemented.\footnote{\citet{HoltSmithElicitation} show that this  mechanism elicits a higher frequency of Bayesian responses than alternative belief elicitation mechanisms.} Subjects make their choices in the same order as in part 2.

\paragraph{Part 4: Eliciting Signal Probabilities}

We elicit the subjects' signal probability assessments $P_{I,i}(\sigma)$ for each of the 10 information structures using multiple choice lists. The choice on each line is between an exogenous probability of winning \$35, and winning the same amount if a ball of a specified color will be drawn from the box that was to be selected at random from the given pair. The computer randomly determines whether a subject is asked to assess the likelihood of a black or a white ball.  Subjects make their choices in the same order as in part 2.

\paragraph{Part 5: Gradual Information Task}
This task is played in four \emph{blocks}. In each of them, subjects see a pair of boxes, each containing some combination of 20 balls (black or white). At the start of the block, the computer randomly selects a box, each with 50\% probability, and shows the subject a sequence of 12 draws \emph{with replacement} from the Selected Box.  After \emph{each} draw, subjects indicated whether they would rather (i) receive \$35 if Box X is the Selected Box (ii) receive \$35 if Box Y is the Selected Box, or (iii) receive \$35 with probability $q$ for some fixed success probability $q$. No choice prematurely ends the sequence of draws, or precludes any future choices. Subjects face, in order, the information structures and values of $q$ in Table \ref{tab:grad_info_choice_problems}.\footnote{These parameter choices imply that a Bayesian decides to bet on a box as soon as he has seen two more balls of one rather than the other color in blocks 1 to 3, and three more balls in block 4.}

\begin{table}[]
\centering
\label{tab:grad_info_choice_problems}
\begin{tabular}{lll}
Block & Information Structure & $q$ \\
\hline
1 & $(0.7,0.7)$           & 0.8 \\
2 & $(0.8,0.8)$           & 0.9 \\
3 & $(0.7,0.7)$           & 0.8 \\
4 & $(0.6,0.6)$           & 0.75 \\ \hline
\end{tabular}
\caption{Choice problems in the Gradual Information Task}
\end{table}


\paragraph{Questionnaire} The final part of the experiment is a questionnaire that includes demographic variables, psychological measures, and the questions from the Cognitive Reflection Test (CRT, \cite{Frederick2005}).


\paragraph{Procedures}
We conducted the experiment at the Stanford Economics Research Laboratory (SERL) using z-tree (\citet{ztree}). Undergraduate and graduate students were recruited from the SERL subject pool database using standard email recruiting procedures. We excluded graduate and undergraduate students in psychology as well as graduate students in economics from participation. A total of 143 individuals participated in 9 sessions of the experiment, in addition to 13 individuals for a pilot session.\footnote{Two sessions presented subjects with the additional information structure $(0.5,0.5)$.  One observation for one subject (out of 1430 total observations) was not recorded due to a database error.}


All sessions were run in May 2013, and each experimental session took between 2 and 2.5 hours to complete, including payment. We distributed and read aloud the instructions directly before each part and informed subjects that their choices in any part would not affect their earnings from any other part. We provided subjects with pen and paper, and neither encouraged nor discouraged their use.  The same two experimenters were present in each session.

Each subject received a \$5 show-up payment, an additional \$15 for completing the experiment, and played for a prize of \$35. In addition, a subject could earn \$1 per CRT question answered correctly.


\paragraph{Conventions}

We code the data such that for every information structure $I$, $P_I(\sigma = 1|s = 1) \geq P_I(\sigma = 0| s = 0)$. Hence, $\sigma = 1$ is the more likely and thus less informative signal realization.

As Figure \ref{fig_agg_valuation}, panel (C) indicates, the information structures in this experiment can be divided into three groups: Information structures S1, S2, S3, and S4 are evenly spaced symmetric information structures. Information structures A1, A2, and A3 are asymmetric information structures that are not on the boundary.  Information structures B1, B2, and B3 are on the boundary.  They afford certainty for one realization of the signal.

Since we have used multiple choice lists in with increments of 3 percentage points, our data are interval coded. Throughout, we use the midpoint of the interval for analysis.

\paragraph{Subject understanding}

Before conducting data analysis, we test whether subjects understood that following the signal is the optimal strategy in the prediction game. In part 1 of the experiment, $83\%$ of our subjects made the correct prediction for all six information structures and just 4 out of 143 subjects (2.8\%) made more than 2 mistakes.\footnote{Moreover, $43.8\%$ of all mistakes occur for the information structure $I=(0.6,0.6)$, the information structure for which such errors are cheapest.} We therefore proceed on the assumption that subjects' valuation of information structures is based on anticipation of this optimal strategy.


\section{Information structure-specific biases}\label{AggregateResults}


How well does the standard model predict average valuations of information structures?  Figure \ref{fig_agg_valuation}, panel (A), plots the mean difference (across subjects) between the elicited and the Bayesian valuation for each information structure.  Information structures are arranged by increasing informativeness.
Perhaps the most salient aspect of Figure \ref{fig_agg_valuation}, panel (A), is the fact that subjects undervalue  all but one information structure, and undervalue them  to a greater extent the more informative they become. In addition, boundary information structures seem to be valued more highly than equally informative information structures in the interior. Meanwhile, there is no indication of a bias relating specifically to asymmetric (but interior) information structures.


\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{../DATA_ANALYSIS/graphs/dev_v_empir_theor.pdf} &
\includegraphics[width = 0.5\textwidth]{../DATA_ANALYSIS/graphs/prefs_beliefs_decomp.pdf} \\
(A) & (B) \\
\multicolumn{2}{c}{\includegraphics[width = 0.5\textwidth]{../DATA_ANALYSIS/graphs/set_of_info_s_idc.pdf}}\\
& \\
\multicolumn{2}{c}{(C)}\\

\end{tabular}
\end{center}
\caption{\label{fig_agg_valuation} Panel (A) depicts the mean deviation of the valuation of an information structure from the Bayesian benchmark, Panel (B) depicts the mean deviation from the subjective beliefs benchmark. Information structures are arranged in order of increasing informativeness. Standard errors are clustered by subject. For reference, panel (C) displays the information structures. Symmetric information structures are labelled $S1$ - $S4$. Asymmetric information structures are $A1$ - $A3$. Boundary information structures are $B1$ - $B3$. For readability, we code the information structures such that $\sigma = 1$ is the weakly more frequent signal for each information structure. In the experiment, there was variation across information structures whether a black or a white ball was the more frequent (and hence less informative) signal realization; see Figure \ref{set_of_info_struct}.}
\end{figure}


Formally, we estimate the model
\bmath \label{aggregateRegression}
v_{I,i} = \beta_0 + \beta_1 v_I^{Bayes} +\beta_2 a_I + \beta_3 b_I + \epsilon_{I,i}
\emath
Here, $a_I = \tfrac{P(\sigma = 1|s=1) - P(\sigma = 0|s=0)}{2}$ is the \emph{asymmetry} of an information structure $I$, measured by its off-diagonal distance from the 45-degree line.\footnote{Recall that we normalized the data such that $a_I \geq 0$ for every information structure $I$.}  $b_I$  is a dummy variable that equals 1 if $I$ is a boundary information structure, and 0 otherwise.  The standard rational agent model requires that $\beta_1 = 1$ and $\beta_0 = \beta_2 = \beta_3 = 0$.


Results are reported in Column 1 of table \ref{tab_results_alt_hypos}.  $\beta_1$ is both significantly less than 1 and significantly larger than 0. Hence, while subjects' valuations increase with informativeness, they do not do so by a sufficient amount. We call this result the \emph{compression effect}.\footnote{Appendix \ref{truncation} shows that the result is robust to controlling for truncation at 1 and 0.5. }  We also find that subjects have a significant preference for information structures where one signal realization removes all uncertainty, on the order of 3.4 percentage points, and hence are subject to a \emph{certainty effect}.\footnote{This result is virtually unchanged when we exclude all individuals who expressed a posterior two or more categories below 100\% in the cases where the Bayesian posterior is 1.} We do not find evidence for asymmetry effects.


Why do subjects undervalue rather than overvalue most information structures? The estimates in Column 1 imply that subjects correctly value information structures with informativeness 0.653 (s.e. 0.024), and overvalue (undervalue) those with lower (higher) informativeness.\footnote{ We calculate this by solving $v = \beta_0 + \beta_1 v$ for $v$, and replacing the parameters $\beta_0$ and $\beta_1$ with the estimates from Column 1.} Accordingly, subjects undervalue most information structures in our experiment because we included a disproportionate number of information structures with comparatively high informativeness (in order to better study asymmetry and boundary effects).


\begin{table}
\begin{center}
\begin{tabular}{lccccccc}

			               	& (1) 		       & (2) 	     	    & (3) 	   	        & (4) 	        	 & (5) 		       & (6) 		        & (7) 		            \\ \hline
Info Structures 		& [S,A,B] 	    	& [S,A,B]		    & [S] 		        & [S,A]		    & [S,B] 		    & [S,B] 		    & [S,A]  		\\
Dependent Var 		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		\\ \hline \hline
$v_I^{Bayes}$ 		&0.603*** 		&0.677***		&0.594*** 	&   0.616***         	&0.655***    	& 0.491***		&0.618***  	\\
			                	&   (0.0427)		&  (0.0386) 		&  (0.0450)		& (0.0439) 		&  (0.0406)		& (0.0427)		&  (0.0437)    		\\
Asymmetry 		       & -0.0237	    & 			           & 		             	& 		             	& 		              	& 			           & -0.0475 		\\
			               	& (0.0260) 		& 	             		& 		            	& 		            	& 		            	& 		              	& (0.0295) 		\\
Boundary 		        	&0.0342***     & 			           & 			            & 		         	& 		              	&0.0273***      	& 		               	\\
				               & (0.00968)   	& 			           & 		              	& 		              	& 			            &  (0.00843)  	& 		           	\\
Constant 		          	&0.260*** 		&0.208*** 		& 0.268***  		& 0.248***  		&0.231*** 		& 0.271***   		&0.249***  		\\
			                	& (0.0326)  		&  (0.0305)		& (0.0338)   	& (0.0326) 		& (0.0323) 		& (0.0330)		&  (0.0326)		\\ \hline
Observations 	     	& 1429  		   & 1429         &  571 	         	& 1000         &  1000 		       &  1000  		        & 1000 		  \\
R-squared 		       &  0.177   		&0.172    		      & 0.186  		& 0.140 	    	&0.202 		&0.207 		&0.141  \\  \hline
\end{tabular}
\end{center}
\caption{\label{tab_results_alt_hypos} Aggregate valuations of information structures. Reported significance levels for $v_I^{Bayes}$ concern the hypothesis that this coefficient is 1. Standard errors clustered by subjects. *** p$<$0.01, ** p$<$0.05, * p$<$0.1}
\end{table}

To be sure that our results are not driven by functional form mis-specification, we check that they are not changed when assessing cleanly comparable subsets of the data.  The compression effect persists when we no longer include $a_I$ and $b_I$ in the regression (Column 2), when we consider only symmetric, evenly spaced information structures (Column 3), when we exclude all boundary information structures (Column 4), and when we exclude all asymmetric non-boundary information structures (Column 5). Certainty effects persist, when we only compare boundary information structures to symmetric information structures (Column 6), and their magnitude is not substantially changed. We do not find significant asymmetry effects, even when we compare only non-boundary asymmetric information structures with symmetric information structures (Column 7).\footnote{To assess the magnitude of the estimated coefficient, note that the information structure $(0.5,1)$ has the maximum possible asymmetry, which is 0.25. This information structure is predicted to be valued only 1.2 percentage points less highly than the equally informative, perfectly symmetric information structure (0.75, 0.75).}


\subsection{Causes}\label{decomposition}

There are at least two channels through which these biases in the demand for information could arise. On the one hand, they could be caused by subjects' anticipation of non-standard belief updating (the \emph{beliefs channel}). On the other hand, subjects could exhibit non-standard information valuations because, even \emph{conditional on their own beliefs}, they have behavioral traits, such as non-standard risk preferences, that affect how they value information, or because they fail to correctly forecast their own belief updating.

To isolate the beliefs channel, we use the data elicited in the Belief Updating Task to derive how a subject values an information structure if the preferences channel is absent, the subject correctly forecasts his belief updating, but belief updating is possibly non-Bayesian. We call this the \emph{subjective-beliefs prediction} of the valuation, and define it by
\bmath \label{eqn_v_pred_emp}
v_{I,i}^{Beliefs} =  P_{I,i}(s=1|\sigma = 1)P_{I,i}(\sigma = 1) + P_{I,i}(s=0|\sigma = 0)P_{I,i}(\sigma = 0)
\emath


To assess the impact of the remaining channels, figure \ref{fig_agg_valuation}, panel (B), compares subjects' valuations of an information structure to the subjective-beliefs prediction. The compression effect seems to disappear, as does the certainty effect. However, average valuations fall significantly short of the predicted valuations, implying that conditional on their own updated beliefs, subjects on average undervalue information.


We formally establish that non-standard belief updating rather than non-standard preferences or a failure to correctly forecast belief updating is the reason for the compression effect and the certainty effect.  Briefly, our approach is the following. Equation (\ref{eqn_v_pred_emp}) predicts how an expected utility maximizer should value an information structure if she correctly forecasts her own belief updating, which, however, may differ from the Bayesian benchmark. If non-standard valuations of information structures derive entirely from non-standard belief updating, then elicited valuations $v_{I,i}$ should not systematically differ from these predictions $v_{I,i}^{Beliefs}$. By contrast, if non-standard preferences play a substantial role (or if subjects fail to correctly anticipate their own belief updating when valuing information structures), we should detect systematic deviations between  $v_{I,i}$ and $v_{I,i}^{Beliefs}$.


A na\"{i}ve testing strategy  would regress $v_{I,i}$ on $v_{I,i}^{Beliefs}$, as well as the characteristics of the information structures, $a_I$, and $b_I$.  The results from this are reported in Column 1 of Table \ref{tab_results_beliefs_values}. Because beliefs, and therefore $v_{I,i}^{Beliefs}$, may be measured with error, however, attenuation bias could cause inconsistent estimates. Consequently, we instrument for the subjective-beliefs prediction of valuations with the Bayesian valuation.\footnote{The identifying assumption is that the correct Bayesian beliefs affect subjects' valuations only \emph{via} their subjective beliefs. This exclusion restriction would not hold if, for instance, subjects were subconsciously influenced by the correct Bayesian posteriors when evaluating information structures, but this influence was not reflected in their \emph{ex post} beliefs. Note that this exercise also establishes that beliefs have a \emph{causal} effect on the demand for information.}  The first-stage regression reported in Column 2 confirms that the instrument is relevant. Column 3 contains the coefficients of interest. The coefficient on $v_{I,i}^{Beliefs}$ is statistically indistinguishable from one ($p = 0.229$), which implies that the compression effect is indeed unrelated to non-standard preferences. The boundary dummy is slightly, but significantly, negative. Hence, the positive certainty effects in the previous section operate mainly through the beliefs channel; the remaining channels add a countervailing effect. Again, there is no evidence of asymmetry effects.


\begin{table}
\begin{center}
\begin{tabular}{lccc}
				& (1) 		& (2) 			& (3) 			\\ \hline
Method                    	& OLS 		& OLS  			& IV 				\\
Dependent Var 		& $v_{I,i}$ 	& $v_{I,i}^{Beliefs}$ 	& $v_{I,i}$ 		\\ \hline \hline
$v_{I,i}^{Beliefs}$ 	& 0.471*** 		& 				&0.984 	\\
                          	& (0.0775) 		& 				& (0.0730)			\\
$v_{I,i}^{Bayes}$ 	&             		&0.613***		&				\\
                           	&            		&  (0.0280) 			&				\\
Asymmetry        	&  -0.0719		& -0.0108		& -0.0131		\\
                          	& (0.0279) 		& (0.0209)  			&  (0.0335) 			\\
Boundary            	&0.0443*** 		&0.0687*** 			& -0.0336**		\\
 				&  (0.0134) 		& (0.0077)		& (0.0141)		\\
Constant            	&0.373***  		&0.276***			& -0.0118			\\
                          	& (0.0612)		&  (0.0229) 			& (0.0540)			\\ \\

Observations 		& 1429 		& 1429  			& 1429			\\
R-squared 		& 0.222	&0.316 			&0.0524			\\ \hline \\
\multicolumn{4}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}
\end{center}
\caption{\label{tab_results_beliefs_values} Relationship between beliefs and valuations. Reported significance levels for $v_I^{Beliefs}$ concern the hypothesis that this coefficient is 1. Standard errors clustered by subjects. }
\end{table}


\section{Individual-specific biases}\label{Heterogeneity}

In this section we study individual-specific biases, and tie them to the information-structure-specific biases. We find that deviations from the Bayesian benchmark are consistent within individuals: a subject whose beliefs move too much upon observing a signal from some information structure tends to update his beliefs too much for all information structures, and by a similar amount.

We employ a one-parameter model that captures much of the variation in behavior across subjects. We estimate, for each individual, their \emph{responsiveness to information} using only data from the belief updating task, and show that  it is significantly correlated with behavior  in two different domains. First, it explains more than 80\% of the non-standard valuation behavior that is attributable to non-standard belief updating. Second, it is significantly correlated with behavior in the Gradual Information Task. This suggests that responsiveness to information is an individual tendency that is stable across different choice environments. It is orthogonal to proxies of mathematical aptitude such as college major and self-reported knowledge of Bayes' law.

Given the estimated parameters, our model also generates the compression effect and the certainty effect that we documented in the preceding section (but predicts magnitudes that fall short of those we observe). This is not an obvious consequence, since these are information-structure-specific effects, and nothing in the responsiveness model directly relates to properties of information structures.


\subsection{Responsiveness to Information}\label{ConsistencyWithin} \label{valuations_across_structural}


For both the information valuation and belief updating tasks, we observe a subject's choices for 10 distinct information structures. If subjects have internally consistent \emph{directional} biases, \emph{within} each of these tasks, a subject's mean deviation from the Bayesian benchmark on two of the information structures should be predictive of her mean deviation on the remaining eight.

Formally, for each information structure $I$ we define subject $i$'s deviation from the Bayesian benchmark by $\Delta  v_{I,i}:= v_{I,i} - v_{I}^{Bayes}$ in the information valuation task and by $\Delta P_{I,i}(s|\sigma) = \logit (P_{I,i}(s|\sigma))  - \logit (P_{I}(s|\sigma))$ in the belief updating task, where $\logit(x) = \log(x / (1 - x))$. We then calculate the extent to which subjects' mean deviation on two information structures correlates with the deviation on the remaining eight, for all possible such splits of the set of information structures, and average. These mean correlations are 0.773 (s.e. 0.035) in the case of $\Delta v_{I,i}$, 0.781 (s.e. 0.035) in the case of $\Delta P_{I,i}(s=1|\sigma = 1)$, and 0.749 (0.075) in the case of $\Delta P_{I,i}(s=0|\sigma = 0)$.\footnote{We control for order and session fixed effects. To perform this exercise with $\Delta P_{I,i}(s=0|\sigma = 0)$ we drop the boundary information structures, and split the sample in pieces of two and five information structures.  This is because $\logit(x)$ is undefined for $x=1$. We bootstrap the standard errors by drawing 1000 bootstrap samples (block-bootstraped on individuals).}  Since all of them are highly significantly positive, we conclude that subjects are internally consistent within tasks.\footnote{This is robust to alternative methods of measuring consistency, such as the Cronbach alpha statistic, which is 0.915, 0.912 and 0.870 for $\Delta v_{I,i}$, $\Delta P_{I,i}(s=1|\sigma = 1)$, and $\Delta P_{I,i}(s= 0|\sigma = 0)$, respectively.  These compare favorably with the standard benchmark of 0.8 (\citet{Kline}), indicating a high consistency of behavior within parts of the experiment.}


This motivates our model of \emph{responsiveness to information}, the extent to which a subjects' beliefs move upon observing information.  This model is adapted from \citet{GretherQJE} and \citet{HoltSmith}.\footnote{These authors did not examine individual consistency.} It explains the individual consistency just documented, and allows us to relate behavior in different parts of the experiment to each other. To formally define responsiveness to information, note that a Bayesian updates beliefs according to the following formula with $\alpha_i$ = 1:
\bmath \label{logit_bayes} \label{alpha_i}
\logit(\underbrace{P_{I,i}(s=1 |\sigma)}_{\text{posterior}}) = \logit(\underbrace{P(s=1)}_{\text{prior}}) +  \alpha_i \underbrace{\log\left( \frac{P_I(\sigma|s=1)}{P_I(\sigma|s=0)}\right)}_{\text{log-likelihood ratio}}
\emath
By letting $\alpha_i$ differ from 1, we allow each subject $i$ to be more or less responsive to information than a Bayesian. We call $\alpha_i$ subject $i$'s \emph{responsiveness to information}.


Our model of responsiveness to information is equivalent to applying a linear-in-log-odds probability weighting function (\citet{GoldsteinEinhorn}, \citet{GonzalesWu}, and \citet{TverskyFox}) to the Bayesian posterior.  Formally, subject $i$'s posterior beliefs and the Bayesian posterior $P_{I}(s=1 |\sigma)$ are related by\footnote{To see this, write equation (\ref{alpha_i}) both for subject $i$ and for a Bayesian, and then substitute out the log-likelihood ratio.}
\bmath \label{probWeighting}
P_{I,i}(s=1|\sigma) = \frac{\delta_i P_I(s=1| \sigma)^{\alpha_i}}{\delta_i P_I(s=1| \sigma)^{\alpha_i} + \big( 1 - P_{I}(s=1 |\sigma)\big)^{\alpha_i}}
\emath
with $\delta_i = \left(\frac{P(1)}{1 - P(1)}\right)^{1 - \alpha_i}$ and $P(1)$ denoting prior beliefs. This function intersects the 45$^{\circ}$-line at the prior $P(1)$, and has the familiar inverse S-shape if $0 < \alpha_i < 1$.


Our model is \emph{not} identical to standard probability weighting (as in \citet{KahnemanTversky1979} or \citet{KahnemanTversky1992}).  Standard theories of probability weighting do not distinguish between lotteries with explicitly stated distributions and lotteries requiring probablistic inference.  Thus, they predict no divergence from Bayesian behavior in the belief updating task.\footnote{In the Belief Updating task, subjects choose between lotteries that pay \$35 with a stated probability and \$0 otherwise, and lotteries that pay \$35 if a given state was realized and \$0 otherwise.  In standard probability weighting, a subject who believes that the given state was realized with probability 70\% is indifferent between a lottery that pays \$35 if that state was realized and an exogenous lottery with a 70\% chance of \$35.}  By contrast, our model involves applying a probability-weighting-style transformation \emph{just to the Bayesian posterior beliefs},  and those divergences from Bayesian behavior are the core of our empirical strategy to predict behavior in other tasks.


\begin{figure}
\begin{center}
\includegraphics[width = 0.6\textwidth]{responsivenessGraph.pdf}
\end{center}
\caption{\label{figResponsiveness} Relation between the posterior $P_{I,i}(s=1|\sigma)$ of a decision maker with responsiveness to information $\alpha_i =0.5$ (plotted on the vertical axis) and the Bayesian posterior $P_{I}(s=1|\sigma)$ (plotted on the horizontal axis) and prior $P_{I}(1)$.}
\end{figure}


We estimate each subject's parameter of responsiveness to information using \emph{only} the data from the Belief Updating Task. We estimate the following regression model using generalized least squares.\footnote{We use GLS rather than OLS, because $\epsilon_{I,i}^1$ and $\epsilon_{I,i}^0$ are plausibly correlated for any given information structure. Hence GLS is a more efficient estimator.}
\bmath \label{SUR}
\left[ \begin{array}{c} \logit(P_{I,i}(s=1|\sigma = 1)) \\ \logit(P_{I,i}(s=0|\sigma = 0)) \end{array} \right] &=&  \alpha_i \left[ \begin{array}{c} \log \Big( \frac{P_I(\sigma = 1|s=1)}{P_I(\sigma = 1|s0)}\Big) \\  \log \Big( \frac{P_I(\sigma = 0|s=0)}{P_I(\sigma = 0|s=1)}\Big)\end{array} \right]  + \left[\begin{array}{c} \epsilon^1_{I,i}\\ \epsilon^0_{I,i}\end{array}\right]
\emath
We do not use data on the boundary information structures because $\logit(P_{I,i}(s=0|\sigma = 0))$ is not defined when $P_{I,i}(s=0|\sigma = 0) = 1$. Logit priors vanish because $P(s=1)=1/2$.\footnote{Responsiveness to information has explanatory power even in the Gradual Information Task, in which subjects generally update from non-uniform priors.}


Figure \ref{fig_hist_alphas} displays the distribution of the estimated parameters $\hat{\alpha}_i$. Our average subject is slightly conservative -- the mean $\hat{\alpha}_i$ is 0.914. The $\hat{\alpha_i}$ are quite variable across subjects, with a standard deviation of 0.396, a minimum of -0.457 and a maximum of 2.191. We reject the null hypothesis that $\hat{\alpha}_i = 1$ at the 5\% level for 43\% (62 of 143) subjects.


\begin{figure}
\begin{center}
\includegraphics[width = 0.6\textwidth]{../DATA_ANALYSIS/graphs/ranking_alphas_all.pdf}
\end{center}
\caption{\label{fig_hist_alphas} The estimated responsiveness parameters $\hat{\alpha}_i$ by subject. Subjects ordered by $\hat{\alpha}_i$.}
\end{figure}

\subsection{ Explanatory power across environments }\label{ConsistencyAcross}\label{AnalysisGradualInfo}

\paragraph{Setting 1: Information Valuation Task}

If responsiveness to information is a \emph{stable} individual trait, then our individual-level estimates of responsiveness should explain behavior in the Information Valuation Task ( an environment which we have \emph{not} used to estimate individual responsiveness parameters ). If responsiveness to information is an \emph{important} trait in our setting, then we should not gain much explanatory power by using the belief updating data in a way that allows for deviations that are idiosyncratic to information structures, rather than by using responsiveness alone.

Subject behavior is correlated across the Information Valuation Task and the Belief Updating Task.  The partial correlation between $\Delta v_{I,i}$ and $\Delta P_{I,i}(s=0|\sigma = 0) + \Delta P_{I,i}(s=1|\sigma = 1)$ (controlling for session and order fixed effects) is 0.328 ($p < 0.0001$).  Hence, a subject's valuations of each information structure are substantially explained by her posterior beliefs for that information structure, \emph{even after we control for informativeness}.

To investigate what part of the variation in information demand responsiveness can explain, we use the data from the Belief Updating Task to construct two distinct predictors of information demand behavior, and compare their explanatory power.

Our \emph{constrained predictor} $\Delta{v}_I(\hat{\alpha}_i)$ uses \emph{only a subject's estimated parameter of responsiveness} to predict how far her valuations deviate from the Bayesian benchmark. Since responsiveness to information is a single parameter, this prevents information-structure-dependent effects, unless they are implied by the responsiveness model. Formally, for each subject and information structure, we use that subject's $\hat{\alpha}_i$ to predict her valuations $\hat{v}_{I}(\hat{\alpha}_i) =P_I(\sigma = 1)\hat{P}_I(s=1|\sigma = 1; \hat{\alpha}_i) + P_I(\sigma = 0)\hat{P}_I(s=0|\sigma = 0; \hat{\alpha}_i)$, where, $\hat{P}_I(s|\sigma; \hat{\alpha}_i)$ are the predicted posteriors. We then define $\Delta{v}_I(\hat{\alpha}_i) := \hat{v}_{I}(\hat{\alpha}_i) - v_I^{Bayes}$.

Our \emph{unconstrained predictor} $\Delta v_{I,i}^{Beliefs}$ uses each subject's posterior beliefs to predict how far her valuation of an information structure will deviate from the Bayesian benchmark \emph{separately for each information structure}.  This permits idiosyncratic deviations for each information structure. Hence, it may be influenced by many distinct belief updating biases along which individuals may be heterogenous, such as different insensitivity to signal characteristics, different rules of thumb, as well as functional form misspecification of the responsiveness model. Formally, $\Delta v_{I,i}^{Beliefs} := v_{I,i}^{Beliefs} - v_I^{Bayes}$.\footnote{Alternatively, one could exclude data from the Signal Probability Assessment task, and construct $v_{I,i}^{Beliefs2} = P(\sigma = 1) P_{I,i}(s=1|\sigma = 1) + P(\sigma = 0)P_{I,i}(s=0|\sigma = 0)$.  This specification uses the \emph{true} signal probabilities and the elicited posterior beliefs.  The results that follow are essentially unchanged if we use this specification instead.}


We compare the explanatory power of the two predictors by estimating
\bmath \label{reg_indiv}
\Delta v_{I,i} = \beta_0 + \beta_1 \Delta v_{I,i}^{Beliefs} + \beta_2 \Delta{v}_I(\hat{\alpha}_i) + \epsilon_i
\emath
We interpret the $R^2$ of this model as the \emph{maximal} part of the variance in the valuations of information structures that can be explained by belief updating. To assess how well responsiveness to information explains this variance, we estimate model (\ref{reg_indiv}) with $\beta_1$ restricted to 0 and study the drop in $R^2$.

The results are in Column 1 and 2 of Table \ref{across_v_reg}.\footnote{Since we use estimates of $\alpha_i$ as explanatory variables, we bootstrap the standard errors. We use the following semiparametric procedure: For each subject we first draw a responsiveness parameter $\alpha_i$ according to the distribution of his estimate of this parameter. We then select a bootstrap sample and perform the estimations. We draw a total of 1000 bootstrap samples.}  The $R^2$ of the unconstrained model is $13.0\%$, which compares to an $R^2$ of $11.3\%$ for the constrained model.\footnote{The difference between these $R^2$ values is significant at all conventional levels. (Testing that the $R^2$ coefficient of the model in column 1 is significantly larger than in the model in column 2 is equivalent to testing that the coefficient on $\Delta v_{I,i}^{Beliefs}$ is zero in the model in column 1, see \citet{Greene}, chapter 5.)}  This shows that our simple one-parameter model captures most ($86.9\%$) of the variation in valuations that can be explained by non-standard belief updating.\footnote{We show in appendix \ref{ResponsivenessSubsetsRobustness} that this is not driven by a particular set of information structures.}  Once we know a subject's general responsiveness to information, a subject's specific \emph{ex post} beliefs contribute little explanatory power. For completeness, Column 3 reports the estimates of model (\ref{reg_indiv}) when $\beta_2$ is set to 0.\footnote{Notably, the R-squared in this instance is lower than that of the \emph{constrained} model.  This suggests that the constrained model exploits underlying structure in the data - the stability of behavior across choice problems in the same task - to produce better estimates.} Our results are not driven by outliers. Columns 4 - 6 show that once we drop the 20\% of subjects with the most extreme estimated parameters of responsiveness to information, the one-parameter model still explains 86.9\% of the variation in valuations that can be explained by non-standard belief updating.


It bears emphasis that the $\Delta{v}_I(\hat{\alpha}_i)$ variables are generated  using only data from the Belief Updating Task. Explanatory power in the Information Valuation Task   obtains even though they are generated without using data from that task.\footnote{Moreover, the $R^2$ coefficients in table \ref{across_v_reg} are deflated by noise in the data. If we average $\Delta v_{I,i}$, $\Delta v_{I,i}^{Beliefs}$, and $\Delta v_{I}(\alpha_i)$ across information structures, we obtain $R^2$ coefficients of 21.6, 16.7 and 20.0 \% in the regressions corresponding to columns 1 - 3 of table \ref{across_v_reg}, respectively.}


\begin{table}

\begin{center}

\begin{tabular}{lccccccccc} \hline
&(1)&(2)&(3)&(4)&(5)&(6)\\
VARIABLE&$\Delta v_{I,i}$&$\Delta v_{I,i}$&$\Delta v_{I,i}$&$\Delta v_{I,i}$&$\Delta v_{I,i}$&$\Delta v_{I,i}$\\  \hline
& & & & & & \\
$\Delta v_{I,i}^{Beliefs}$ &0.246*** &         &0.413*** &0.214*** &       \\
&  (0.066)&         &(0.083)  &  (0.080)&         &  (0.092) \\
$\Delta v_{I}(\hat{\alpha_i})$ &0.300*** &0.550*** &         &1.020    &    1.244&    \\
&  (0.146)&  (0.144)&         &  (0.359)&(0.360)  &    \\
Constant 	&-0.038***&-0.035***&-0.044***&-0.039***&-0.036***&-0.049***\\
&  (0.008)&  (0.008)&(0.008)  &  (0.008)&  (0.008)&  (0.009)\\
&         &         &         &         &         &         \\
Obs. 				& 1,429 & 1,429 & 1,429 & 1,119 & 1,119 & 1,119 \\
$R^2$ 				& 0.130 & 0.113 & 0.116 & 0.107 & 0.093 & 0.056  \\
 10\% trim & - & - & - & Yes & Yes & Yes \\

 \hline
\end{tabular}
\end{center}

\caption{\label{across_v_reg} Explaining and decomposing deviations from benchmark valuations. Significance levels on the slope parameters concern the hypothesis that the coefficients are 1. Standard errors are bootstrapped using the following semiparametric procedure: For each subject we first draw a responsiveness parameter $\alpha_i$ according to the distribution of his estimate of this parameter. We then select a bootstrap sample and perform the estimations. We repeat this 1000 times.  *** p$<$0.01, ** p$<$0.05, * p$<$0.1.}
\end{table}


\paragraph{Setting 2: Gradual Information Task}

The Gradual Information Task is a second environment that allows us to test the robustness of responsiveness to information. Subjects observed a \emph{sequence} of informative signals about the state of the world, and made choices that revealed when their certainty about the underlying state crossed a given threshold. Hence while the Belief Updating Task reveals how certain a subject is about the state of the world after observing a signal realization, the Gradual Information Task reveals how much information a subject needs to observe before attaining a given level of certainty. Additionally, and unlike in the Belief Updating Task, subjects in the Gradual Information Task (i) generally update from non-symmetric priors (after having observed the first piece of information), (ii) make aggregative assessments from a long sequence of signals, (iii) make ternary decisions rather than an essentially continuous choice, (iv) choose after having observed information rather than conditional on observing information in the future.


Consider how a subject perfectly described by the responsiveness model would behave in the Gradual Information Task. Her posterior belief $p_{I,i}^t$ that  $s = 1$ after having observed $t$ conditionally independent signal realizations is derived by repeated application of equation (\ref{alpha_i}).  Thus, her logit-beliefs thus follow a random walk with drift, where the responsiveness parameter $\alpha_i$ scales the rate of movement.  Hence, we expect individuals who are \emph{less} responsive to information to require \emph{more} ball draws before they are willing to bet on a box, and vice versa.  Figure \ref{beliefstream} illustrates.


\begin{figure}
\begin{center}
\includegraphics[width = .8\textwidth]{gradInfoPic.pdf}
\end{center}
\caption{\label{beliefstream}Evolution of Bayesian beliefs and those of a subject with responsiveness $\alpha_i<1$. The subject requires more information than the Bayesian before her beliefs cross $q_i$.}
\end{figure}


Our variable of interest is how much more (or less) sure than $q_I$ a Bayesian would be when at the point in time $\tau_{I,i}$ at which subject $i$ first bets on a box. We define $\Delta b_{I,i} = \logit(b_{I,i}^{\tau_{I,i}}) - \logit(q_I)$ where $b_{I,i}^{\tau_{I,i}}$ is the Bayesian posterior at $\tau_{I,i}$ for the sequence of information that subject $i$ has observed.\footnote{Alternatively, one could consider the number of balls that a subject saw before betting on a box.  Since ball draws are conditionally i.i.d., however, this variable is noisy. Applying the logit transformation to beliefs makes our measure comparable across different information structures and ensures that overly and underly responsive subjects receive the same weight in the regression.}  This is marked on Figure \ref{beliefstream}.  The responsiveness model predicts that $\Delta b_{I,i}$ is negatively related to $\alpha_i$.

We regress $\Delta b_{I,i}$ on each subject's responsiveness $\hat{\alpha}_i$,  which is estimated using only data from the Belief Updating Task.  Columns 1 and 2 in Table \ref{table_grad_info_all} report the results.\footnote{Our data are truncated, since the random drawing of balls caused some subjects to not observe a sufficient amount of information that would have made them willing to bet on the state of the world. Consequently, we use Tobit regressions.}  Responsiveness has the expected effect on subject performance in the Gradual Information Task.  This effect is significant with and without the inclusion of demographic controls.

To take a fully structural approach, we take each subject's responsiveness parameter, and generate a prediction for $\Delta b_{I}$ given the sequence of signals each subject saw.  We call this variable $\Delta b_{I}(\hat{\alpha_i})$.  We then regress measured $\Delta b_{I,i}$ on the prediction $\Delta b_{I}(\hat{\alpha_i})$.  Columns 3 and 4 in Table \ref{table_grad_info_all} report the results.  The coefficient has the expected sign, but is not statistically significant.   This suggests that, while measured responsiveness predicts behavior in different choice problems, the functional form we assumed in Equation \ref{alpha_i} may not straightforwardly carry over to the Gradual Information Task.


\begin{table}
\begin{center}
\begin{small}
\begin{tabular}{lcccc}
\hline
						& (1) 			& (2) 			& (3) 			& (4) 			\\
VARIABLES  				&$\Delta b_{I,i}$	&  $\Delta b_{I,i}$	& $\Delta b_{I,i}$ 	& $\Delta b_{I,i}$ 	\\ \hline
						&  				&  				&  				&  				\\
$\hat{\alpha}_i$					&-1.65** 			& -1.77**	\\
						&(0.79)			& (0.75)	\\
$\Delta b_I (\hat{\alpha}_i)$   	&				& 				& 0.23 			& 0.22 			\\
						&				& 				& (0.19)			& (0.18)			\\
Constant  					& 4.03			&3.80*** 			&2.28*** 			&2.08*** 			\\
 						& (0.86)			& (1.00)			& (0.38)			& (0.68)			\\
Demographic controls		& No				& Yes			& No				& Yes			\\
Observations 				& 572 			& 572 			& 572 			& 572 			\\
Pseudo-$R^2$ 				& 0.028 			& 0.032 			& 0.014 			& 0.013 			\\
\hline
\multicolumn{5}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}
\end{small}
\end{center}
\caption{\label{table_grad_info_all} Predicting behavior in the Gradual Information Task. Results are from Tobit-regressions. Demographic controls include race and gender. For goodness-of-fit, we report the \citet{McKelveyZavoina} pseudo-$R^2$.}
\end{table}


\paragraph{Alternative Explanations}
We have elicited subjects' posterior beliefs and valuations of information structures using multiple price lists. These are a specific implementation of the \citet*{BDM} mechanism, which experimental subjects sometimes find hard to understand (e.g. \citet{AndersenMPL} and \citet{Beauchampetal.}).

If there is consistent individual heterogeneity in subjects' biases in this elicitation procedure, those potentially confound the effects of responsiveness to information.  We address this concern in two ways. First, if our results were an artifact of consistent individual heterogeneity in misunderstandings of multiple price list procedures, we would not expect to find that our estimates of responsiveness are correlated with behavior in the gradual information task, which does not involve multiple price lists.

Second, regarding consistency between the belief updating and information valuation tasks, our design lets us difference out the effect of consistent individual heterogeneity in the misunderstanding of multiple price lists. The reason is that misunderstanding of multiple price lists may affect behavior in all three tasks that rely on multiple price lists, specifically the information valuation, belief updating, and probability assessment tasks. Responsiveness to information, by contrast, may affect behavior only if belief updating plays a role; specifically in the belief updating and information valuation tasks. Statistically significant and substantial correlations remain, showing that individual consistency does not (only) derive from misunderstandings of the multiple price lists. We defer the formal analysis to Appendix \ref{extremenessAvoidance}.\footnote{ Another potential explanation for individual consistency concerns the different random draws of balls that different subjects were exposed to in part 1 of the experiment. These varying experiences could simultaneously affect all choices relating to random ball draws, including the probability assessment task. Accordingly, we can difference out such effects. The consistency in behavior across the belief updating and information valuation tasks remains. See Appendix \ref{extremenessAvoidance} for details.}


\subsection{Correlation with other variables} \label{CorrelatesAlphaMain} \label{correlations}

In a cognitively demanding experiment such as ours, behavior may be caused by more general stable individual traits such as mathematical aptitude and cognitive style. While we do not have an \textit{ex ante} hypothesis whether more mathematically proficient subjects would be more or less responsive to information, one might expect that they would differ from the Bayesian benchmark to a smaller extent.  Our data do not suggest any such correlation.


\begin{table}
\begin{center}
\begin{tabular}{lcccc}
 & (1) & (2) & (3) & (4) \\
VARIABLES & $\alpha_i$ & $\alpha_i$  & $|\alpha_i-1|$  & $|\alpha_i-1|$ \\ \hline
 &  &  &  &  \\
Knows Bayes' law 	 				& 0.098 & -0.021 & 0.013 & 0.003 \\
 				 				& (0.081) & (0.053) & (0.065) & (0.026) \\
CRT score  		 				& 0.051 & 0.047* & -0.039 & -0.027** \\
  				 				& (0.038) & (0.025) & (0.031) & (0.012) \\
STEM major 		 				& -0.044 & -0.008 & -0.053 & -0.021 \\
 				 				& (0.078) & (0.050) & (0.063) & (0.025) \\
Business or economics major  			& -0.142 & -0.075 & 0.016 & 0.005 \\
 				 				& (0.133) & (0.084) & (0.107) & (0.043) \\


Demographic controls & Yes & Yes & Yes & Yes \\
Method & OLS & QR & OLS & QR\\
&&&&\\
Constant & 0.892*** & 0.785*** & 0.449*** & 0.294*** \\
 & (0.131) & (0.081) & (0.105) & (0.041) \\
 &  &  &  &  \\
Observations & 143 & 143 & 143 & 143 \\
 R-squared & 0.143 & - & 0.155 & - \\ \hline
 \multicolumn{5}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}
\end{center}
\caption{\label{correlatesAlpha} Cognitive style, mathematical aptitude and responsiveness to information. The excluded category for college major includes all non-STEM fields that are neither business nor economics. Columns 1 and 3 list the results of OLS regressions, columns 2 and 4 those of median regressions. All models estimated with session and order fixed effects.}
\end{table}


Formally, we regress both the estimated parameters of responsiveness and their deviation from the Bayesian benchmark on subjects' self-reported knowledge of Bayes' law, their cognitive style as measured by the CRT score, their college major, and a vector of demographic characteristics consisting of dummies for race and gender.

Column 1 of Table \ref{correlatesAlpha} shows that neither of our measures of mathematical aptitude and cognitive style is significantly associated with responsiveness to information. Once we use median regression instead of least squares (Column 2), we find a significantly positive relation between CRT scores and the subjects' responsiveness. We reach similar conclusions by associating these variables with the extent to which a subject deviates from the Bayesian benchmark. Using OLS, we find no significant relations (Column 3), but once we estimate median regressions,  we discover that subjects who score one point higher in the CRT test are in closer alignment with the Bayesian benchmark by 2.6 percentage points (Column 4).\footnote{In line with these findings, \citet{StanovichWest} conclude that for the type of belief updating problems as we study, discrepancies between normative models and measured behavior are not easily explained by recourse to cognitive capacity.}


\subsection{Responsiveness and information-structure-specific biases}\label{responsivenessAggregate}

We now tie our one-parameter model to the information-structure-specific biases that we documented in section \ref{AggregateResults}. Such a relation is perhaps surprising -- after all, the responsiveness model is set up to capture individual-level biases that do not depend on the specifics of information structures.

The model predicts that under-responsive subjects will exhibit both the compression effect and the certainty effect.  An under-responsive subject still correctly identifies wholly uninformative signals.  However, as a signal becomes marginally more informative, an under-responsive subject's beliefs move \emph{less} than a Bayesian's would, which means that his valuations also increase less than Bayesian's.  This generates the compression effect.  The certainty effect arises because a subject also has correct beliefs upon observing a perfectly informative signal realization. Therefore, for a boundary information structure, the individual will have too low a posterior only for the signal realization that leaves residual uncertainty, but not for the one that reveals the state of the world. For an interior information structure, by contrast, the individual will have too low a posterior regardless of the signal realization.

\begin{table}
\setlength{\tabcolsep}{4pt}
\begin{center}
\begin{tabular}{lcccccccc} \hline
 & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)\\
VARIABLES & $v_I(\alpha_i)$ & $v_{I,i}$ & $v_I(\alpha_i)$ & $v_{I,i}$ & $v_I(\alpha_i)$ & $v_{I,i}$ &$v_I(\alpha_i)$ & $v_{I,i}$ \\
Sample & all & all & all & all & $\alpha_i < 1$  & $\alpha_i < 1$   & $\alpha_i > 1$  & $\alpha_i > 1$  \\
 \hline
 &  &  &  &  &  &  \\
$v_I^{Bayes}$ & 0.883*** & 0.603*** & 0.877*** & 0.615*** & 0.751*** & 0.560*** & 1.020*** & 0.677*** \\
 & (0.026) & (0.043) & (0.027) & (0.041) & (0.046) & (0.060) & (0.003) & (0.054) \\
$a_I$ & -0.007** & -0.024 & 0.023*** & -0.090 & 0.030*** & -0.060 & 0.013*** & -0.126 \\
 & (0.003) & (0.026) & (0.005) & (0.075) & (0.008) & (0.105) & (0.002) & (0.106) \\
$(a_I)^2$  &  &  & -0.084*** & 0.190 & -0.177*** & -0.069 & 0.023*** & 0.485 \\
 &  &  & (0.020) & (0.198) & (0.033) & (0.248) & (0.002) & (0.315) \\
$b_I$ & 0.022*** & 0.034*** & 0.023*** & 0.032*** & 0.056*** & 0.060*** & -0.015*** & 0.002 \\
 & (0.006) & (0.010) & (0.006) & (0.010) & (0.009) & (0.015) & (0.002) & (0.012) \\
Constant & 0.062*** & 0.260*** & 0.065*** & 0.252*** & 0.115*** & 0.253*** & 0.010** & 0.251*** \\
 & (0.013) & (0.033) & (0.013) & (0.031) & (0.023) & (0.044) & (0.004) & (0.044) \\
 &  &  &  &  &  &  &  &  \\
Observations & 1,429 & 1,429 & 1,429 & 1,429 & 760 & 760 & 669 & 669 \\
 $R^2$ & 0.507 & 0.177 & 0.508 & 0.177 & 0.436 & 0.179 & 0.925 & 0.220 \\ \hline
\end{tabular}
\end{center}
\caption{\label{tabResponsivenessAggregate} Relating the responsiveness model to the information-structure-specific effects. Reported significance levels on $v_I^{Bayes}$ concern the hypothesis that the coefficient is 1. Standard errors clustered by subject.  *** p$<$0.01, ** p$<$0.05, * p$<$0.1}
\end{table}


Formally, we derive these predictions from the probability-weighting interpretation of the responsiveness model when $0<\alpha_i < 1$. The compression effect is due to the fact that the slope of (\ref{probWeighting}) is lower than 1 for all Bayesian posteriors that are not too close to 0 or 1. Moreover, this function is convex to the right of the prior, and most strongly so for Bayesian posteriors close to 1. Hence it implies the certainty effect (as well as a more general but weak preference for asymmetric information structures).


Since the majority of our subjects in our sample are less responsive than a Bayesian, we expect that the responsiveness model with the estimated parameters will reproduce the information-structure-specific biases. For each subject $i$ and information structure $I$ we use the estimated parameter $\alpha_i$ to predict the distance between that subjects' valuation of that information structure and the Bayesian benchmark. We then estimate model (\ref{aggregateRegression}) with the predicted deviations as dependent variable.  Table \ref{tabResponsivenessAggregate}, column 1 reports the results. For comparison, column 2 reproduces the analysis with the measured deviations as dependent variable. The coefficient on $v_I^{Bayes}$ is significantly smaller than 1, and hence the model captures the compression effect. The extent of this effect, however, is significantly smaller than in column 2. The predicted valuations also exhibit a significant certainty effect. Again, the extent of the bias in the predicted data is smaller than the measured one.


The coefficient on asymmetry is negative even for the predicted valuations, in contrast to the theoretical predictions. This indicates that the linear functional form may be overly restrictive. Thus we include squared asymmetry as a regressor and report the results in columns 3 and 4. Estimates of the compression and certainty effects do not substantially change, neither for measured, nor for predicted valuations.


The mechanics that link the responsiveness model to the information-structure-specific biases rest on the fact that the average subject is less responsive than a Bayesian. For overly responsive subjects, the information-structure-specific biases are predicted to reverse in sign. To test this hypothesis, we reproduce the above analysis  separately on the subsamples of subjects with $\alpha_i<1$ and those with $\alpha_i>1$. Columns 5 - 8 display the results. (The dependent variable is the predicted deviation in columns 5 and 7, and the measured deviation in columns 6 and 8.)  While we do not see the predicted reversal of the compression and certainty effects for overly responsive subjects we do find that both the compression effect and the certainty effect are significantly stronger for the subjects with $\alpha_i<1$ than for the remaining subjects ($p < 0.01$ for each of the effects).


\section{Conclusion}\label{conclusion}

Economists study many situations where agents learn about their environment by acquiring costly stochastic information. In this article, we examine information demand behavior and its relation to belief updating and non-standard risk preferences across a range of choice contexts.

We find two biases that depend on the properties of information structures. First, subjects generally underreact to increases in the informativeness of an informative signal.  Second, they value boundary information structures disproportionately highly. These two biases do not persist, however, once we condition on subjects' elicited beliefs, rather than the objective Bayesian posteriors. Hence, these biases are  primarily the result of non-standard belief updating. They are unlikely caused by non-standard risk preferences.


Importantly, we identify a bias that is specific to individuals. They differ \emph{consistently} in responsiveness to information, the extent to which their beliefs move upon observing information. Individual-level estimates of responsiveness to information have explanatory power in two different choice problems. They explain about 80\% of the variation in information demand that is maximally attributable to belief updating. Responsiveness is unrelated to proxies for mathematical aptitude.  The one-parameter model of responsiveness to information we estimate is also able to generate the compression and certainty effects, biases that depend on the properties of information structures. Intuitively, this arises because our average subject is less responsive than a Bayesian, and such subjects' beliefs move too little for signals of intermediate informativeness, but are in line with the benchmark for perfectly informative and for perfectly uninformative signals.


If we use our individual estimates of responsiveness in a structural model, however, they fail to predict behavior in our sequential belief updating task with statistical significance (although the raw coefficients are significantly correlated with behavior in that task, and even the estimates from the structural model attain the predicted sign). This opens interesting questions for further research. Specifically, the raw parameters are more correlated with behavior in that task than the structural predictions that rely on them.  This suggests that the structural model is \textit{correctly identifying} a dimension of individual heterogeneity, but \textit{incorrectly predicting} how this heterogeneity manifests in different tasks.  In particular, if a single signal and a sequence of signals contain the same information, the responsiveness model constrains agents to evaluate both identically.  However, it could instead be that individuals over-infer from short sequences of signals, but under-infer from long ones \citep{GriffinTversky, KraemerWeber}, so a model that predicts well across different kinds of belief-updating tasks may have to abandon this invariance.

More generally, the finding highlights the importance of determining the scope of applicability of models of belief updating.  Two seemingly opposite conclusions emerge from prior work on belief updating.  An early psychology literature employed balls-and-urn tasks, and documented conservatism, \textit{i.e.} that subjects update too little from signals (see \citet{PetersonBeach} for a review).  The heuristics and biases approach, by contrast, has often relied on tasks that do not provide enough formal structure to allow for the explicit calculation of Bayesian posterior beliefs, and has often documented base-rate neglect, which leads subjects to update too much from weak signals (for instance the Tom W. example in \citet{KahnemanTverskyRepresentativeness}).  In our data, we find that subjects value signals of varying informativeness as though they were more alike, compared to the predictions of the standard model (the compression effect).  An expanded model that exploits this observation might accommodate both conservatism and base-rate neglect.


Moreover, the tasks in our experiment are all based on a similar balls-and-urns setting. It would be valuable for future research to explore how predictive responsiveness to information is across more radically different environments. For instance, are subjects' choices in our experimental tasks related to the extent to which they update upon receiving the results of a medical test, or of the extent to which they change their financial investments upon learning financial news? Recent research suggests that such an investigation holds promise. On the one hand, \citet{Buseretal.Updating} find that responsiveness to information predicts tournament entry. On the other hand, that the geopolitical forecasters studied in  \citet{AugenblickRabin} exhibit individual consistency in the volatility of belief streams.\footnote{To see the connection between responsiveness and volatility, consider two agents who see the same stream of information about a binary event. Suppose one of them is extremely responsive whereas the other's responsiveness is close to zero. The former agent's beliefs will tend to jump forth and back between values close to 0 and close to 1, whereas the latter agent's beliefs will remain close to his prior.}

Responsiveness to information also bears potential for theoretical generalization. Our application studies the two-states case, in which posterior variance is a deterministic function of the posterior mean.  In the more general case, does responsiveness operate on the dimension of means, variance, or both? That is, in the language of \citet{MooreHealy}, is an overly responsive individual prone to overplacement, to overprecision, or to both?

Once individual stability and predictiveness is further established, and more responsive individuals can be identified, economists may begin using the trait to better target programs such as those fostering technology adoption in developing countries. Relatedly, in settings where non-Bayesian updating is costly, such as medical decision making, subjects who are excessively or insufficiently responsive to information might be identified and given help to bring their decision-making in line with the normative benchmark.


\newpage


\ifx\undefined\bysame
\newcommand{\bysame}{\leavevmode\hbox to\leftmargin{\hrulefill\,\,}}
\fi
\begin{thebibliography}{xx}

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\newpage
\begin{appendix}

\setcounter{page}{0}

\begin{center}
{\Huge \textbf{ONLINE-APPENDIX}}\\
\vspace{0.5cm}
{\Huge \textbf{NOT FOR PUBLICATION}}
\end{center}
\newpage


\section{Additional Correlates of Responsiveness to Information}\label{CorrelatesResponsiveness}

The last stage of our experiment was a questionnaire to elicit a range of demographic and psychological measures from our subjects. Here, we extend the analysis of section \ref{correlations} to include these psychological measures. We administered the substantive questions of the life-orientation-test-revised (\citet{LOTR}) as a measure of optimism, and we elicited agreement\footnote{The possible responses were ``I agree a lot'', ``I agree a little'', ``I disagree a little'', and ``I disagree a lot''.} to a number of statements intended to capture superstition,\footnote{We opted not to use the standard superstition questionnaires such as \citet{TobacykMilford} because the statements in these tests seemed too blunt given our subject pool. Instead we elicited the level of agreement with the following three statements: ``I own an object that I feel brings me good luck during exams'', ``I am uncomfortable if I am assigned seat number 13 in a theatre'', and ``I refrain from taking risks when I feel I am having an unlucky day''.} spirituality,\footnote{We elicited the level of agreement to the statements ``Today, I am less spiritual than I was 10 years ago'', ``My religious beliefs are roughly the same as they were when I was 12 years old'', and ``Over the past ten years, I have become more spiritual''.} and political orientation.\footnote{The questions were ``In political elections, I vote (if you cannot vote in the US, please answer who you would vote for if you could)'' with answers ``always republican'', ``usually republican'', ``about equally often republican as democrat'', ``usually democrat'', and ``always democrat''.} We interspersed these latter questions with the LOT-R questions and with each other. We added some of these questions only after having run the first two sessions of the experiment. The variables optimism, superstition and religiosity are elicited using several statements each. We use the first principal component as a single index for each of these variables. As in section \ref{correlations}, we use both OLS and median regressions.

Table \ref{corr_alphas} reports the results. The inclusion of psychological variables does not alter our conclusions regarding the effect of cognitive style and proxies for mathematical aptitude. Psychological variables are largely insignificant. An exception is the finding that more optimistic subjects deviate from the Bayesian benchmark to a slightly (1 - 2 percentage points) lesser extent. Moreover, while more religious subjects appear to deviate from the Bayesian benchmark to a larger extent when we consider the OLS estimates, this effect disappears in the median regression, and hence appears to be driven by outliers.

Demographic variables have greater explanatory power. While males are not, on average, more or less responsive than females, they tend to be closer to the Bayesian benchmark; by a substantial amount (4 to 10 percentage points). Additionally, Caucasians and Asians are more responsive to information and deviate from the Bayesian benchmark to a lesser extent than the excluded category \emph{other}, whereas these effects are reversed for African-Americans.


\begin{table}

\begin{center}
\begin{tabular}{lcccc} \hline
 & (1) & (2) & (3) & (4) \\
VARIABLES & $\hat{\alpha}_i$  & $\hat{\alpha}_i$ & $|\hat{\alpha}_i-1|$  & $|\hat{\alpha}_i-1|$ \\ \hline
 &  &  &    \\
Knows Bayes' law 					& 0.0857 & 0.0257 & -0.00442 & -0.0134		\\
 								& (0.0935) & (0.0732) & (0.0765) & (0.0365) 	\\
Male 							& 0.0124 & -0.00219 & -0.0414 & -0.0922*** 	\\
 								& (0.0793) & (0.0634) & (0.0649) & (0.0320) 	\\
Science or engineering major 			& 0.0107 & 0.0152 & -0.0131 & 0.0106 		\\
 								& (0.0874) & (0.0714) & (0.0715) & (0.0342)	\\
Business or economics major 			& -0.0323 & -0.00808 & 0.0487 & 0.0366		\\
 								& (0.161) & (0.123) & (0.132) & (0.0651) 		\\
African-American					& -0.143 & -0.273* & 0.0303 & 0.163** 		\\
								& (0.186) & (0.149) & (0.152) & (0.0749) 		\\
Asian 							& 0.0867 & 0.187* & -0.132 & -0.118***		\\
 								& (0.115) & (0.0950) & (0.0944) & (0.0448) 	\\
Caucasian						& 0.255** & 0.243*** & -0.184** & -0.100**	\\
								& (0.104) & (0.0854) & (0.0855) & (0.0432)	\\
CRT 								& 0.0397 & 0.0395 & -0.0236 & 0.000602		\\
 								& (0.0446) & (0.0358) & (0.0365) & (0.0182)	\\
LOTR 							& 0.0134 & 0.0235 & -0.0145 & -0.0178** 		\\
 								& (0.0234) & (0.0189) & (0.0191) & (0.00880) 	\\
Superstition 					 	& 0.00863 & -0.0289 & 0.000100 & 0.000535	\\
 								& (0.0352) & (0.0266) & (0.0288) & (0.0137)	\\
Religiosity 						& -0.0134 & 0.0101 & 0.0592** & 0.00998		\\
 								& (0.0356) & (0.0273) & (0.0291) & (0.0142)	\\
Constant							& 0.506*** & 0.641*** & 0.470*** & 0.288***	\\
								& (0.172) & (0.130) & (0.141) & (0.0667)		\\
Method		& OLS & MR & OLS & MR	\\
 							& 	 &  &   \\
Observations & 120 & 120 & 120 & 120 \\
 R-squared & 0.171 & - & 0.199 & - \\ \hline

\multicolumn{5}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}

\end{center}
\caption{\label{corr_alphas} Responsiveness to information, demographic and psychological variables. The excluded category for major is \emph{humanities and social sciences}. The excluded category for race is \emph{other}. Columns 1 and 3 display OLS estimates. Columns 2 and 4  display estimates from median regressions. All models include session and order fixed effects. }
\end{table}


\section{Robustness Checks}


\subsection{Truncation in the aggregate analysis} \label{truncation}

The compression effect might be a statistical artifact: First, it is impossible to enter valuations that exceed one, and second, even for subjects who do not know how to value a particular information structure, it might be obvious that stating valuations short of 0.5 is suboptimal. Consequently, large upward errors are possible for less informative information structures, but not for more informative ones, and the opposite holds for large downward errors.

We show that the compression effect is not a statistical artifact, as follows. If for information structure $I$, the maximal upwards deviation from the theoretical valuation is $M$, then we set all valuations with $v_{I,i}<v_I^{Bayes}- M$ equal to $v_I^{Bayes}-M$. Similarly, if the maximal downwards deviation from the theoretical valuation is $m$, then we set all valuations with $v_{I,i}>v_I^{Bayes}+m$ equal to $v_I^{Bayes}+m$. We perform the analysis of section \ref{AggregateResults} on this data. If the compression effect is purely a statistical artifact, then the coefficient on $v_I^{Bayes}$ will be equal to 1 in this procedure.  On the other hand, if the true coefficient on $v_I^{Bayes}$ is less than 1, then this procedure will bias that coefficient estimate upwards, and bias the other coefficients in ways that are difficult to sign.

 Table \ref{tab_results_alt_hypos_trunc} displays the results. We find that the compression effect is reduced, but remains substantial and highly statistically significant, and hence is not a statistical artifact.

\begin{table}

\begin{center}
\begin{tabular}{lccccccc}

			               	& (1) 		       & (2) 	     	    & (3) 	   	        & (4) 	        	 & (5) 		       & (6) 		        & (7) 		            \\ \hline
Info Structures 		& [S,A,B] 	    	& [S,A,B]		    & [S] 		        & [S,A]		    & [S,B] 		    & [S,B] 		    & [S,A]  		\\
Dependent Var 		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		& $v_{I,i}$		\\ \hline \hline
$v_I^{Bayes}$ 		&0.866*** 		&0.945**		&0.867*** 	&   0.880***         	&0.919***    	& 0.876***		&0.884***  	\\
			                	&   (0.026)		&  (0.025) 		&  (0.025)		& (0.027) 		&  (0.024)		& (0.026)		&  (0.027)    		\\
Asymmetry 		       & -0.065***    & 			           & 		             	& 		             	& 		              	& 			           & -0.093*** 		\\
			               	& (0.024) 		& 	             		& 		            	& 		            	& 		            	& 		              	& (0.028) 		\\
Boundary 		        	&0.041***     & 			           & 			            & 		         	& 		              	&0.018***      	& 		               	\\
				               & (0.007)   	& 			           & 		              	& 		              	& 			            &  (0.005)  	& 		           	\\
Constant 		          	&0.072*** 		&0.013 		& 0.075***  		& 0.057** 		&0.042* 		& 0.068***   		&0.060**  		\\
			                	& (0.023)  		&  (0.024)		& (0.023)   	& (0.024) 		& (0.023) 		& (0.023)		&  (0.024)		\\ \hline
Observations 	     	& 1429  		   & 1429         &  571 	         	& 1000         &  1000 		       &  1000  		        & 1000 		  \\
R-squared 		       & 0.444    		& 0.436    		      & 0.503  		& 0.350 	    	&0.549 		&0.552 		&0.355   		\\ \\

\multicolumn{8}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}
\end{center}

\caption{\label{tab_results_alt_hypos_trunc} Results of section \ref{AggregateResults} correcting for truncation. Standard errors clustered by subjects. Reported significance levels for $v_I^{Bayes}$ concern the hypothesis that this coefficient is 1.}
\end{table}


\subsection{Effect of different urn sizes}\label{appendixUrnSize}

 One of the urns in our experiment contains half as many balls as the other. This design choice prevents simple non-Bayesian heuristics from coinciding with Bayesian behavior. The difference in the size of the urns, however, might by itself cause artifacts.\footnote{We thank an anonymous referee for this suggestion.} To address this concern, we regress the valuations $v_{I,i}$ on the properties of the information structure, with the inclusion of a control variable \emph{small urn} that encodes whether the more informative signal realization is associated with the smaller or with the larger box. This variable equals  1, -1, and 0 for information structures below, above, and on the diagonal in Figure \ref{set_of_info_struct} (B), respectively.\footnote{Consequently, the information structures labelled 6 and 3 in Figure 1 (B) are assigned value 1, and the information structures labelled 5, 7, 8, 9 in that figure are assigned value -1.}


\begin{table}


\begin{center}
\setlength{\tabcolsep}{3pt}

\begin{tabular}{lcccccccc} \hline
 & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) \\
VARIABLES & \multicolumn{2}{c}{$v_{I,i}$} &  \multicolumn{2}{c}{$\Delta v_{I,i}$}&  \multicolumn{2}{c}{$\Delta P_{I,i}(s = 1 | \sigma = 1)$} &  \multicolumn{2}{c}{$\Delta P_{I,i}(s = 0 | \sigma = 0)$} \\ \hline
 &  &  &  &  &  &  &  &  \\
Informativeness & 0.603*** & 0.622*** & -0.396*** & -0.374*** & -0.299*** & -0.271*** & -0.441*** & -0.435*** \\
 & (0.043) & (0.043) & (0.043) & (0.043) & (0.032) & (0.035) & (0.033) & (0.036) \\
Asymmetry & -0.024 & -0.001 & -0.026 & -0.001 & 0.103*** & 0.136*** & -0.198*** & -0.191*** \\
 & (0.026) & (0.032) & (0.026) & (0.032) & (0.024) & (0.028) & (0.027) & (0.027) \\
Boundary & 0.034*** & 0.033*** & 0.034*** & 0.032*** & 0.049*** & 0.048*** & 0.076*** & 0.076*** \\
 & (0.010) & (0.010) & (0.010) & (0.010) & (0.008) & (0.008) & (0.011) & (0.011) \\
Small Urn &  & -0.007 &  & -0.008* &  & -0.010*** &  & -0.002 \\
 &  & (0.005) &  & (0.005) &  & (0.004) &  & (0.004) \\
Constant & 0.260*** & 0.240*** & 0.259*** & 0.238*** & 0.199*** & 0.171*** & 0.328*** & 0.322*** \\
 & (0.033) & (0.033) & (0.033) & (0.033) & (0.026) & (0.030) & (0.027) & (0.030) \\
 &  &  &  &  &  &  &  &  \\
Observations & 1,429 & 1,429 & 1,412 & 1,412 & 1,412 & 1,412 & 1,412 & 1,412 \\
 R-squared & 0.177 & 0.178 & 0.052 & 0.053 & 0.068 & 0.069 & 0.080 & 0.080 \\ \hline
\end{tabular}

\end{center}

\caption{\label{smallUrn} Robustness to differing urn sizes. \emph{Small Urn} is a dummy variable that equals 1, -1, and 0 for information structures below, above, and on the diagonal in Figure \ref{set_of_info_struct} (B), respectively. Standard errors clustered by subject. *** p$<$0.01, ** p$<$0.05, * p$<$0.1}
\end{table}

Column 2 of Table \ref{smallUrn} displays the result (Column 1 replicates the regression without the inclusion of the control variable). The  coefficient on the variable \emph{small urn} is very small and not significantly different from zero. The only notable effect of including this variable is to further attenuate the coefficient on asymmetry which was insignificant anyway. The remaining coefficients are not substantially affected.

Additionally, we can perform a similar exercise using the deviation from the Bayesian valuation, and the deviation in Bayesian posteriors as dependent variables. Columns 3 - 8 display the results. There is no statistically significant effect of \emph{small urn}, except on the extent to which $P_{I,i}(s = 1 | \sigma = 1)$ deviates from the Bayesian benchmark. Even that effect is very small, and does not substantially alter the estimated coefficients associated with the other right-hand-side variables.

We conclude that the difference in urn sizes does not substantially affect our results.


\subsection{ Other factors that could cause within-subject consistency}\label{extremenessAvoidance}


To address the potential confounds emanating from misunderstandings of the multiple price lists, we use deviations in the probability assessment task, as well as the realization of ball draws a subject has observed in stage 1 of the experiment to predict the deviations in the belief updating task that can be explained by these factors. Formally, we estimate the following model
\bmath \label{predictioneq1}
\Delta {x}_{I,i} = \beta_{0,I}^x + \sum_{I'=1}^{10} \Big[ \beta_{I',I}^x \Delta {P^1_{i,I'}(\sigma = 1)} +  \gamma_{I',I}^x \Delta {P^0_{i,I'}(\sigma = 1)} \Big]+  \sum_{k=1}^6 c^k_i+ \epsilon_{I,i} ^{x}
\emath
where $\Delta P^{\sigma'}_{i,I'}(\sigma = 1) = P^{\sigma'}_{i,I'}(\sigma = 1)  - P_{I'}(\sigma = 1)$. Here, $P^{1}_{i,I'}(\sigma = 1)$ is subject $i$'s elicited  probability of receiving signal $\sigma = 1$ if she was directly asked about that signal, and $P^{0}_{i,I'}(\sigma = 1)$ is the implied subjective probability  of  receiving signal $\sigma = 1$ if she was asked about signal $\sigma = 0$.\footnote{We randomly elicited either the probabilities of white or black balls. Hence either $\Delta P_{i,I'}(\sigma = 1) = 0$ or $\Delta P_{i,I'}(\sigma = 0) = 0$ for each observation $(i,I)$. We allow the parameters in the first stage regression to differ depending on whether we elicited the probability of a black ball or the probability of a white ball to allow for the greatest generality.}  Moreover, $c_i^k$ is the color of the ball that subject $i$ saw in round $k$ of part 1, the $\epsilon_{I,i}^x$ are assumed to be i.i.d. with mean zero and $x\in \{P(s=1|\sigma = 1), P(s=0|\sigma = 0)\}$ are elicited posterior beliefs. We include session and order fixed-effects.

This model is the most general model with the properties that (i) the functional form relating behavior in the Probability Assessment Task to behavior in a given other part of the experiment is the same for all subjects  (although its parameters may vary across subjects)  and (ii) this function is linear. In particular, it allows for each $\Delta {x}_{I,i}$ to depend differentially on all of the $\Delta {P_{i,I'}(\sigma)}$, for $I, I' \in \{1, \ldots, 10\}$, $\sigma \in \{0, 1\}$. Thus it accommodates explanations such as people placing a constant premium (heterogenous across individuals) on the exogenous lottery relative to the evaluation option or ideas such as some people being more prone to avoid extreme lines in the multiple choice lists.


To assess consistency across parts of the experiment, we obtain the residuals and predicted values from regression (\ref{predictioneq1}). Because we will use these variables on the right hand side, we average across information structures, to alleviate attenuation bias. Moreover, for the sake of interpretability, we normalize all variables by their standard deviation. We denote the resulting variables by $\hat{\epsilon}_{i} ^{x}$ and $\hat{x}_{i}$, respectively, for $x\in \{P(s=1|\sigma = 1), P(s=0|\sigma = 0)\}$.\footnote{We exclude the boundary information structures when averaging over information structures.}

We then regress the deviation from the Bayesian valuation in the information valuation task (normalized by its standard deviation) jointly on the predicted deviation and the residual in the belief updating stage. Specifically, we estimate
\bmath \label{effect}
\Delta x_{I,i} = \gamma_0 + \gamma_1 \hat{\epsilon}_{I,i} ^{P(s=1|\sigma = 1)} + \gamma_2 \hat{\epsilon}_{I,i} ^{P(s=0|\sigma = 0)}  + \eta_{I,i}
\emath
If deviations in valuations are entirely an experimental artifact (such misunderstandings of the multiple price lists), we expect the residuals to have no explanatory power, and hence, $\gamma_1 = \gamma_2 = 0$. If, on the other hand, deviations are entirely due to heterogenous responsiveness to information, we expect the residuals to have significant explanatory power, $\gamma_1 >0, \gamma_2 > 0$.

We estimate equation (\ref{effect}) using session fixed effects, clustering standard errors by subject. We find that the coefficients on the residuals are $\gamma_1 = 0.106$ (s.e. 0.058) and $\gamma_2 = 0.143$ (s.e. 0.051). The p-value of a test for joint significance is below 0.01.


Hence, even controlling for the effects mentioned above, deviations from the Bayesian benchmark in the Belief Updating Task are still highly significant predictors of deviations from the Bayesian benchmark in the Information Valuation Task.


\subsection{Responsiveness and subsets of information structures} \label{ResponsivenessSubsetsRobustness}

In section \ref{ConsistencyAcross} we found that our constrained predictor explains 86.6\% of the variation in the deviations from Bayesian valuations that can maximally be explained using beliefs. We check that this result is not driven by a small number of information structures.

We consider the variation in the deviations from Bayesian valuations that can maximally be explained using beliefs separately for all the symmetric, asymmetric, and boundary information structures, and their respective complements. We also consider the subsets consisting of the two, four, and eight information structures that are most extreme in terms of informativeness.\footnote{We do not consider the subset of the six most extreme information structures, as this would require us to break ties for equally informative information structures.}

Table \ref{responsivenessSubsets} shows that for just two of these subsets the fraction of variance explained by the unconstrained predictor drops substantially below 70\%. In both of these cases, the subset of information structures is small. We conclude that our finding in section \ref{ConsistencyAcross} is not driven by a small number of information structures.


\begin{table}

\begin{center}
\begin{tabular}{cccc}

Subset 					& $R^2_{constrained}/R^2_{both}$ 	& Subset 	& $R^2_{constrained}/R^2_{both}$ \\ \hline

$S$						& 	79.0\%	& $A, B$					&	93.0\%	\\
$A$						& 	98.5\%	& $S, B$					&	73.3\%	\\
$B$						& 	55.8\%	& $S, A$					&	91.7\%	\\
$S_1, B_3$				&	26.1\%	& $S_2, S_3, B_1,B_2, A$ 	&	96.9\%	\\
$S_1, S_2, B_2,B_3$		&	67.9	\%	& $S_2, S_3, B_1,B_2, A$ 	&	96.9\%	\\
$S_1, S_2, S_4, A, B_2,B_3$	&	88.2	\%	& $S_3, B_1$				&	78.5\%	\\ \hline
\end{tabular}
\end{center}

\caption{\label{responsivenessSubsets} Fraction of explainable variance in individual biases in valuations that can be explained using responsiveness alone, on subsets of information structures. $R^2_{constrained}$ is the $R^2$ of a regression of $\Delta v_{I,i}$ on the constrained predictor. $R^2_{both}$ is the $R^2$ of a regression of $\Delta v_{I,i}$ on both the constrained and the unconstrained predictor.}
\end{table}


\section{Responsiveness to Information and Risk Aversion}\label{model}

Here we demonstrate that the predictions of heterogeneity in responsiveness to information are not collinear with those of heterogenous risk aversion.

Suppose there is a binary state of the world $s \in \{0,1\}$ with a known prior $p = P(s = 1)$. An agent can take either a risk free act $C$ that provides a state independent payoff $c$, or he can opt for a risky act $R$ which has downside $d$ in state $s = 0$ and upside $u$ in state $s = 1$, where $d < c < u$, as depicted in the following table.

\begin{center}
\begin{tabular}{c|cc|}
Act & $s = 0 $ & $ s = 1 $ \\ \hline
$C$ & $c$ & $c$ \\
$R$ & $d$ & $u$ \\ \hline
\end{tabular}
\end{center}

Before making a decision, the agent has the opportunity to acquire a costly informative signal $\sigma\in \{0, 1\}$ about the state of the world. We are interested in the comparative statics of the agent's willingness to pay for the information with respect to (i) his responsiveness to information and (ii) his risk aversion.

It is clear that the agent's willingness to pay for information is increasing in his responsiveness to information, regardless of which act he would choose in the absence of information. This is because an agent who is more responsive to information experiences a greater reduction in subjective uncertainty upon observing any signal.

Importantly, however, the comparative statics with respect to risk aversion depend on what action the agent would take in the absence of information.

First, suppose that the parameters of the problem are such that in the absence of information the agent chooses the risky act $R$. In this case, a signal realization either increases the subjective probability for $s = 1$, in which case the agent also decides for the risky act after observing the signal. Or it decreases the subjective probability for $s = 1$, in which case the agent might opt for the riskless act $C$ instead. Hence, information can prevent the agent from deciding for $R$ when doing so would likely pay the downside $d$.\footnote{Conditional on this signal, $EU(R)<EU(C)$.} An increase in risk aversion decreases the attractiveness of $R$ for any posterior about $s$, but leaves the attractiveness of $C$ unchanged. Therefore, a more risk averse agent who, in the absence of information, opts for the risky option has more to gain from information. Hence, his willingness to pay for information \emph{increases} with risk aversion.

Second, suppose that the parameters of the problem are such that in absence of information, the agent would instead optimally choose the riskless act $C$. Now, if the signal realization  decreases the subjective probability that $s = 1$, then the agent still opts for the riskless act $C$. If it increases the subjective probability that $s = 1$, however, the agent might choose the risky act $R$ instead. Hence, information can help the agent to take the risk $R$ when doing so is likely to pay the upside $u$.\footnote{Conditional on this signal, $EU(R)>EU(C)$.} Now, an increase in risk aversion decreases the attractiveness of the risky option $R$ relative to the riskless option $C$. Hence, a more risk averse agent has less to gain from information. Consequently, if in the absence of information the agent opts for the riskless option, an increase in risk aversion \emph{decreases} willingness to pay for information.\footnote{We can use this simple model to derive predictions of individual behavior across markets for information. Suppose that market $A$ provides information for decisions in which people by default opt for the risky option, whereas market $B$ provides information for decisions in which people usually opt for the riskless option. If people are all similarly risk averse but heterogenous in responsiveness to information, then individuals' willingness to pay for information is positively correlated across the two markets. On the other hand, if people are all similarly responsive to information but differ in risk aversion, we expect negative correlation between individual willingness to pay across markets $A$ and $B$.}


\newpage
\section{Experimental Instructions}

\includepdf[pages = 1 - 16]{instructions.pdf}

\end{appendix}

\end{document}
